ÎïÀíѧ½Ì³Ì£¨µÚ¶þ°æ£©ÉϲáÏ°Ìâ´ð°¸
µÚÒ»Õ ÖʵãÔ˶¯Ñ§
1 -1 Öʵã×÷ÇúÏßÔ˶¯,ÔÚʱ¿Ìt ÖʵãµÄλʸΪr,ËÙ¶ÈΪv ,ËÙÂÊΪv,t ÖÁ(t £«¦¤t)ʱ¼äÄÚµÄλÒÆΪ¦¤r, ·³ÌΪ¦¤s, λʸ´óСµÄ±ä»¯Á¿Îª¦¤r ( »ò³Æ¦¤£ür£ü),ƽ¾ùËÙ¶ÈΪv,ƽ¾ùËÙÂÊΪv£® (1) ¸ù¾ÝÉÏÊöÇé¿ö,Ôò±ØÓÐ( ) (A) £ü¦¤r£ü= ¦¤s = ¦¤r
(B) £ü¦¤r£ü¡Ù ¦¤s ¡Ù ¦¤r,µ±¦¤t¡ú0 ʱÓУüdr£ü= ds ¡Ù dr (C) £ü¦¤r£ü¡Ù ¦¤r ¡Ù ¦¤s,µ±¦¤t¡ú0 ʱÓУüdr£ü= dr ¡Ù ds (D) £ü¦¤r£ü¡Ù ¦¤s ¡Ù ¦¤r,µ±¦¤t¡ú0 ʱÓУüdr£ü= dr = ds (2) ¸ù¾ÝÉÏÊöÇé¿ö,Ôò±ØÓÐ( )
(A) £üv£ü= v,£üv£ü= v (B) £üv£ü¡Ùv,£üv£ü¡Ù v (C) £üv£ü= v,£üv£ü¡Ù v (D) £üv£ü¡Ùv,£üv£ü= v
·ÖÎöÓë½â (1) ÖʵãÔÚt ÖÁ(t £«¦¤t)ʱ¼äÄÚÑØÇúÏß´ÓP µãÔ˶¯µ½P¡äµã,¸÷Á¿¹ØϵÈçͼËùʾ, ÆäÖз³Ì¦¤s £½PP¡ä, λÒÆ´óС£ü¦¤r£ü£½PP¡ä,¶ø¦¤r £½£ür£ü-£ür£ü±íʾÖʵãλʸ´óСµÄ±ä»¯Á¿,Èý¸öÁ¿µÄÎïÀíº¬Ò岻ͬ,ÔÚÇúÏßÔ˶¯ÖдóСҲ²»ÏàµÈ(×¢£ºÔÚÖ±ÏßÔ˶¯ÖÐÓÐÏàµÈµÄ¿ÉÄÜ)£®µ«µ±¦¤t¡ú0 ʱ,µãP¡äÎÞÏÞÇ÷½üPµã,ÔòÓУüdr£ü£½ds,µ«È´²»µÈÓÚdr£®¹ÊÑ¡(B)£® (2) ÓÉÓÚ£ü¦¤r £ü¡Ù¦¤s,¹Ê
¦¤r¦¤s,¼´£üv£ü¡Ùv£® ?¦¤t¦¤tµ«ÓÉÓÚ£üdr£ü£½ds,¹Ê
drds?,¼´£üv£ü£½v£®Óɴ˿ɼû,Ӧѡ(C)£® dtdt1 -2
dr(1)
dtÒ»Ô˶¯ÖʵãÔÚij˲ʱλÓÚλʸr(x,y)µÄ¶Ëµã´¦,¶ÔÆäËٶȵĴóСÓÐËÄÖÖÒâ¼û,¼´
£» (2)
drdt£»
ds(3)
dt£» (4)
?dx??dy???????dt??dt?22£®
ÏÂÊöÅжÏÕýÈ·µÄÊÇ( )
(A) Ö»ÓÐ(1)(2)ÕýÈ· (B) Ö»ÓÐ(2)ÕýÈ·
1
(C) Ö»ÓÐ(2)(3)ÕýÈ· (D) Ö»ÓÐ(3)(4)ÕýÈ· ·ÖÎöÓë½â
drdt±íʾÖʵ㵽×ø±êÔµãµÄ¾àÀëËæʱ¼äµÄ±ä»¯ÂÊ,ÔÚ¼«×ø±êϵÖно¶ÏòËÙÂÊ£®Í¨³£Ó÷ûºÅvr±íʾ,
ÕâÊÇËÙ¶ÈʸÁ¿ÔÚλʸ·½ÏòÉϵÄÒ»¸ö·ÖÁ¿£»drdt±íʾËÙ¶ÈʸÁ¿£»ÔÚ×ÔÈ»×ø±êϵÖÐËٶȴóС¿ÉÓù«Ê½v22?ds¼Ædt?dx??dy?Ëã,ÔÚÖ±½Ç×ø±êϵÖÐÔò¿ÉÓɹ«Ê½v???????dt??dt?Çó½â£®¹ÊÑ¡(D)£®
1 -3 Öʵã×÷ÇúÏßÔ˶¯,r ±íʾλÖÃʸÁ¿, v±íʾËÙ¶È,a±íʾ¼ÓËÙ¶È,s ±íʾ·³Ì, a£ô±íʾÇÐÏò¼ÓËٶȣ®¶ÔÏÂÁбí´ïʽ,¼´
(1)d v /dt £½a£»(2)dr/dt £½v£»(3)ds/dt £½v£»(4)d v /dt£ü£½a£ô£® ÏÂÊöÅжÏÕýÈ·µÄÊÇ( )
(A) Ö»ÓÐ(1)¡¢(4)ÊÇ¶ÔµÄ (B) Ö»ÓÐ(2)¡¢(4)ÊÇ¶ÔµÄ (C) Ö»ÓÐ(2)ÊÇ¶ÔµÄ (D) Ö»ÓÐ(3)ÊǶԵÄ
dv±íʾÇÐÏò¼ÓËÙ¶Èa,Ëü±íʾËٶȴóСËæʱ¼äµÄ±ä»¯ÂÊ,ÊǼÓËÙ¶ÈʸÁ¿ÑØËٶȷ½ÏòµÄÒ»¸ö·ÖÁ¿,dtdrdsÆð¸Ä±äËٶȴóСµÄ×÷Óã»ÔÚ¼«×ø±êϵÖбíʾ¾¶ÏòËÙÂÊvr(ÈçÌâ1 -2 ËùÊö)£»ÔÚ×ÔÈ»×ø±êϵÖбíʾÖÊ
dtdt·ÖÎöÓë½â
£ô
µãµÄËÙÂÊv£»¶ø
dvdt±íʾ¼ÓËٶȵĴóС¶ø²»ÊÇÇÐÏò¼ÓËÙ¶Èa£ô£®Òò´ËÖ»ÓÐ(3) ʽ±í´ïÊÇÕýÈ·µÄ£®¹ÊÑ¡(D)£®
1 -4 Ò»¸öÖʵãÔÚ×öÔ²ÖÜÔ˶¯Ê±,ÔòÓÐ( ) (A) ÇÐÏò¼ÓËÙ¶ÈÒ»¶¨¸Ä±ä,·¨Ïò¼ÓËÙ¶ÈÒ²¸Ä±ä (B) ÇÐÏò¼ÓËٶȿÉÄܲ»±ä,·¨Ïò¼ÓËÙ¶ÈÒ»¶¨¸Ä±ä (C) ÇÐÏò¼ÓËٶȿÉÄܲ»±ä,·¨Ïò¼ÓËٶȲ»±ä (D) ÇÐÏò¼ÓËÙ¶ÈÒ»¶¨¸Ä±ä,·¨Ïò¼ÓËٶȲ»±ä
·ÖÎöÓë½â ¼ÓËٶȵÄÇÐÏò·ÖÁ¿a£ôÆð¸Ä±äËٶȴóСµÄ×÷ÓÃ,¶ø·¨Ïò·ÖÁ¿anÆð¸Ä±äËٶȷ½ÏòµÄ×÷Óã®Öʵã×÷Ô²ÖÜÔ˶¯Ê±,ÓÉÓÚËٶȷ½Ïò²»¶Ï¸Ä±ä,ÏàÓ¦·¨Ïò¼ÓËٶȵķ½ÏòÒ²ÔÚ²»¶Ï¸Ä±ä,Òò¶ø·¨Ïò¼ÓËÙ¶ÈÊÇÒ»¶¨¸Ä±äµÄ£®ÖÁÓÚ
a£ôÊÇ·ñ¸Ä±ä,ÔòÒªÊÓÖʵãµÄËÙÂÊÇé¿ö¶ø¶¨£®Öʵã×÷ÔÈËÙÂÊÔ²ÖÜÔ˶¯Ê±, a£ôºãΪÁ㣻Öʵã×÷ÔȱäËÙÂÊÔ²ÖÜÔ˶¯
ʱ, a£ôΪһ²»ÎªÁãµÄºãÁ¿,µ±a£ô¸Ä±äʱ,ÖʵãÔò×÷Ò»°ãµÄ±äËÙÂÊÔ²ÖÜÔ˶¯£®Óɴ˿ɼû,Ӧѡ(B)£® 1 -5 ÒÑÖªÖʵãÑØx Öá×÷Ö±ÏßÔ˶¯,ÆäÔ˶¯·½³ÌΪs£®Çó£º
(1) ÖʵãÔÚÔ˶¯¿ªÊ¼ºó4.0 sÄÚµÄλÒƵĴóС£» (2) ÖʵãÔÚ¸Ãʱ¼äÄÚËùͨ¹ýµÄ·³Ì£» (3) t£½4 sʱÖʵãµÄËٶȺͼÓËٶȣ®
·ÖÎö λÒƺÍ·³ÌÊÇÁ½¸öÍêÈ«²»Í¬µÄ¸ÅÄֻÓе±Öʵã×÷Ö±ÏßÔ˶¯ÇÒÔ˶¯·½Ïò²»¸Ä±äʱ,λÒƵĴóС²Å»áÓë·³ÌÏàµÈ£®ÖʵãÔÚt ʱ¼äÄÚµÄλÒƦ¤x µÄ´óС¿ÉÖ±½ÓÓÉÔ˶¯·½³ÌµÃµ½£º¦¤xx?2?6t2?2t3,ʽÖÐx µÄµ¥Î»Îªm,t µÄµ¥Î»Îª
?xt?x0,¶øÔÚÇó·³Ìʱ,¾Í±Ø
Ðë×¢Òâµ½ÖʵãÔÚÔ˶¯¹ý³ÌÖпÉÄܸıäÔ˶¯·½Ïò,´Ëʱ,λÒƵĴóСºÍ·³Ì¾Í²»Í¬ÁË£®Îª´Ë,Ðè¸ù¾Ý
dx?0À´dtÈ·¶¨ÆäÔ˶¯·½Ïò¸Ä±äµÄʱ¿Ìtp ,Çó³ö0¡«tp ºÍtp¡«t ÄÚµÄλÒÆ´óС¦¤x1 ¡¢¦¤x2 ,Ôòt ʱ¼äÄڵķ³Ì
2
dxd2xs??x1??x2,ÈçͼËùʾ,ÖÁÓÚt £½4.0 s ʱÖʵãËٶȺͼÓËٶȿÉÓúÍ2Á½Ê½¼ÆË㣮
dtdt
Ìâ 1-5 ͼ
½â (1) ÖʵãÔÚ4.0 sÄÚλÒƵĴóС
¦¤x?x4?x0??32m
(2) ÓÉ µÃÖªÖʵãµÄ»»Ïòʱ¿ÌΪ
dx?0 dttp?2s (t£½0²»ºÏÌâÒâ)
Ôò
¦¤x1?x2?x0?8.0m
¦¤x2?x4?x2??40m
ËùÒÔ,ÖʵãÔÚ4.0 sʱ¼ä¼ä¸ôÄڵķ³ÌΪ
s?¦¤x1?¦¤x2?48m
(3) t£½4.0 sʱ
v?dx??48m?s?1
dtt?4.0sd2xa?2??36m.s?2
dtt?4.0s1 -6 ÒÑÖªÖʵãµÄÔ˶¯·½³ÌΪr(1) ÖʵãµÄÔ˶¯¹ì¼££»
(2) t £½0 ¼°t £½2£óʱ,ÖʵãµÄλʸ£»
(3) ÓÉt £½0 µ½t £½2£óÄÚÖʵãµÄλÒƦ¤r ºÍ¾¶ÏòÔöÁ¿¦¤r£»
·ÖÎö ÖʵãµÄ¹ì¼£·½³ÌΪy £½f(x),¿ÉÓÉÔ˶¯·½³ÌµÄÁ½¸ö·ÖÁ¿Ê½x(t)ºÍy(t)ÖÐÏûÈ¥t ¼´¿ÉµÃµ½£®¶ÔÓÚr¡¢¦¤r¡¢¦¤r¡¢¦¤s À´Ëµ,ÎïÀíº¬Ò岻ͬ,(Ïê¼ûÌâ1-1·ÖÎö). ½â (1) ÓÉx(t)ºÍy(t)ÖÐÏûÈ¥t ºóµÃÖʵã¹ì¼£·½³ÌΪ
?2ti?(2?t2)j,ʽÖÐr µÄµ¥Î»Îªm,t µÄµ¥Î»Îª£ó£®Çó£º
y?2?ÕâÊÇÒ»¸öÅ×ÎïÏß·½³Ì,¹ì¼£Èçͼ(a)Ëùʾ£®
12x 4(2) ½«t £½0£óºÍt £½2£ó·Ö±ð´úÈëÔ˶¯·½³Ì,¿ÉµÃÏàӦλʸ·Ö±ðΪ
r0?2j , r2?4i?2j
ͼ(a)ÖеÄP¡¢Q Á½µã,¼´Îªt £½0£óºÍt £½2£óʱÖʵãËùÔÚλÖã® (3) ÓÉλÒƱí´ïʽ,µÃ
3
¦¤r?r2?r1?(x2?x0)i?(y2?y0)j?4i?2j
ÆäÖÐλÒÆ´óС
¦¤r?(¦¤x)2?(¦¤y)2?5.66m
2222?¦¤r?r2?r0?x2?y2?x0?y0?2.47m
¶ø¾¶ÏòÔöÁ¿¦¤r
Ìâ 1-6 ͼ
1 -7 ÖʵãµÄÔ˶¯·½³ÌΪ
x??10t?30t2
y?15t?20t2
ʽÖÐx,y µÄµ¥Î»Îªm,t µÄµ¥Î»Îª£ó£®
ÊÔÇó£º(1) ³õËٶȵĴóСºÍ·½Ïò£»(2) ¼ÓËٶȵĴóСºÍ·½Ïò£®
·ÖÎö ÓÉÔ˶¯·½³ÌµÄ·ÖÁ¿Ê½¿É·Ö±ðÇó³öËٶȡ¢¼ÓËٶȵķÖÁ¿,ÔÙÓÉÔ˶¯ºÏ³ÉËã³öËٶȺͼÓËٶȵĴóСºÍ·½Ïò£®
½â (1) ËٶȵķÖÁ¿Ê½Îª
vx?dx??10?60t dtdyvy??15?40t
dt-1
µ±t £½0 ʱ, v0x £½-10 m¡¤£ó , v0y £½15 m¡¤£ó ,Ôò³õËٶȴóСΪ
-1
v0?v0x?v0y?18.0m?s?1
Éèv0Óëx ÖáµÄ¼Ð½ÇΪ¦Á,Ôò
22tan¦Á?v0yv0x??3 2¦Á£½123¡ã41¡ä
(2) ¼ÓËٶȵķÖÁ¿Ê½Îª
ax?Ôò¼ÓËٶȵĴóСΪ
dvydvx??40m?s?2 ?60m?s?2 , ay?dtdta?ax?ay?72.1m?s?2
Éèa Óëx ÖáµÄ¼Ð½ÇΪ¦Â,Ôò
4
22
tan¦Â?ayax??2 3-1
¦Â£½-33¡ã41¡ä(»ò326¡ã19¡ä)
1 -8 Ò»Éý½µ»úÒÔ¼ÓËÙ¶È1.22 m¡¤£óÉÏÉý,µ±ÉÏÉýËÙ¶ÈΪ2.44 m¡¤£óʱ,ÓÐÒ»ÂÝË¿×ÔÉý½µ»úµÄÌ컨°åÉÏËÉÍÑ,Ì컨°åÓëÉý½µ»úµÄµ×ÃæÏà¾à2.74 m£®¼ÆË㣺(1)ÂÝË¿´ÓÌ컨°åÂäµ½µ×ÃæËùÐèÒªµÄʱ¼ä£»(2)ÂÝË¿Ïà¶ÔÉý½µ»úÍâ¹Ì¶¨Öù×ÓµÄϽµ¾àÀ룮
·ÖÎö ÔÚÉý½µ»úÓëÂÝË¿Ö®¼äÓÐÏà¶ÔÔ˶¯µÄÇé¿öÏÂ,Ò»ÖÖ´¦Àí·½·¨ÊÇÈ¡µØÃæΪ²Î¿¼Ïµ,·Ö±ðÌÖÂÛÉý½µ»úÊúÖ±ÏòÉϵÄÔȼÓËÙ¶ÈÔ˶¯ºÍ³õËÙ²»ÎªÁãµÄÂÝË¿µÄ×ÔÓÉÂäÌåÔ˶¯,ÁгöÕâÁ½ÖÖÔ˶¯ÔÚͬһ×ø±êϵÖеÄÔ˶¯·½³Ìy1 £½
-2
y1(t)ºÍy2 £½y2(t),²¢¿¼ÂÇËüÃÇÏàÓö,¼´Î»Ê¸ÏàͬÕâÒ»Ìõ¼þ,ÎÊÌâ¼´¿É½â£»ÁíÒ»ÖÖ·½·¨ÊÇÈ¡Éý½µ»ú(»òÂÝË¿)Ϊ
²Î¿¼Ïµ,Õâʱ,ÂÝË¿(»òÉý½µ»ú)Ïà¶ÔËü×÷ÔȼÓËÙÔ˶¯,µ«ÊÇ,´Ë¼ÓËÙ¶ÈÓ¦¸ÃÊÇÏà¶Ô¼ÓËٶȣ®Éý½µ»úÏáµÄ¸ß¶È¾ÍÊÇÂÝË¿(»òÉý½µ»ú)Ô˶¯µÄ·³Ì£®
½â1 (1) ÒÔµØÃæΪ²Î¿¼Ïµ,È¡ÈçͼËùʾµÄ×ø±êϵ,Éý½µ»úÓëÂÝË¿µÄÔ˶¯·½³Ì·Ö±ðΪ
1y1?v0t?at2
21y2?h?v0t?gt2
2µ±ÂÝË¿ÂäÖÁµ×Ãæʱ,ÓÐy1 £½y2 ,¼´
11v0t?at2?h?v0t?gt2
22t?2h?0.705s
g?a12gt?0.716m 2 (2) ÂÝË¿Ïà¶ÔÉý½µ»úÍâ¹Ì¶¨Öù×ÓϽµµÄ¾àÀëΪ
d?h?y2??v0t?½â2 (1)ÒÔÉý½µ»úΪ²Î¿¼Ïµ,´Ëʱ,ÂÝË¿Ïà¶ÔËüµÄ¼ÓËٶȴóСa¡ä£½g £«a,ÂÝË¿ÂäÖÁµ×Ãæʱ,ÓÐ
10?h?(g?a)t2
2t?(2) ÓÉÓÚÉý½µ»úÔÚt ʱ¼äÄÚÉÏÉýµÄ¸ß¶ÈΪ
2h?0.705s
g?a1h??v0t?at2
2Ôò d?h?h??0.716m
5
Ìâ 1-8 ͼ
1 -9 ÖʵãÑØÖ±ÏßÔ˶¯,¼ÓËÙ¶Èa£½4 -t ,ʽÖÐaµÄµ¥Î»Îªm¡¤£ó ,tµÄµ¥Î»Îª£ó£®Èç¹ûµ±t £½3£óʱ,x£½9 m,v £½2 m¡¤£ó ,ÇóÖʵãµÄÔ˶¯·½³Ì£®
·ÖÎö ±¾ÌâÊôÓÚÔ˶¯Ñ§µÚ¶þÀàÎÊÌâ,¼´ÒÑÖª¼ÓËÙ¶ÈÇóËٶȺÍÔ˶¯·½³Ì,±ØÐëÔÚ¸ø¶¨Ìõ¼þÏÂÓûý·Ö·½·¨½â¾ö£®ÓÉa-1
2
-2
?dvdxºÍv?¿ÉµÃdv?adtºÍdx?vdt£®Èça£½a(t)»òv £½v(t),Ôò¿ÉÁ½±ßÖ±½Ó»ý·Ö£®Èçdtdt¹ûa »òv²»ÊÇʱ¼ät µÄÏÔº¯Êý,ÔòÓ¦¾¹ýÖîÈç·ÖÀë±äÁ¿»ò±äÁ¿´ú»»µÈÊýѧ²Ù×÷ºóÔÙ×ö»ý·Ö£® ½â ÓÉ·ÖÎöÖª,Ó¦ÓÐ
?
ÓÉ
vv0dv??adt
0tµÃ v1?4t?t3?v0 (1)
3?xx0dx??vdt
0tµÃ x½«t£½3£óʱ,x£½9 m,v£½2 m¡¤£ó´úÈë(1)¡¢(2)µÃ
-1
?2t2?14t?v0t?x0 (2) 12v0£½-1 m¡¤£ó-1, x0£½0.75 m
ÓÚÊǿɵÃÖʵãÔ˶¯·½³ÌΪ
x?2t2?A¡¢B ΪÕýºãÁ¿,Çóʯ×ÓÏÂÂäµÄËٶȺÍÔ˶¯·½³Ì£®
14t?0.75 121 -10 һʯ×Ó´Ó¿ÕÖÐÓɾ²Ö¹ÏÂÂä,ÓÉÓÚ¿ÕÆø×èÁ¦,ʯ×Ó²¢·Ç×÷×ÔÓÉÂäÌåÔ˶¯,ÏÖ²âµÃÆä¼ÓËÙ¶Èa£½A -Bv,ʽÖÐ
½â ѡȡʯ×ÓÏÂÂä·½ÏòΪy ÖáÕýÏò,ÏÂÂäÆðµãΪ×ø±êԵ㣮
(1) ÓÉÌâÒâÖª aÓ÷ÖÀë±äÁ¿·¨°Ñʽ(1)¸ÄдΪ
?dv?A?Bv (1) dtdv?dt (2)
A?Bv½«Ê½(2)Á½±ß»ý·Ö²¢¿¼ÂdzõʼÌõ¼þ,ÓÐ
6
tdvdv??dt
0A?Bv?µÃʯ×ÓËÙ¶È vÓÉ´Ë¿ÉÖªµ±,t¡ú¡Þʱ,v(2) ÔÙÓÉvvv0?A(1?e?Bt) B?AΪһ³£Á¿,ͨ³£³ÆΪ¼«ÏÞËٶȻòÊÕβËٶȣ® B?dyA?(1?e?Bt)²¢¿¼ÂdzõʼÌõ¼þÓÐ dtBytA?Btdy?(1?e)dt ?0?0By?AAt?2(e?Bt?1) BB-2
µÃʯ×ÓÔ˶¯·½³Ì
1 -11 Ò»Öʵã¾ßÓк㶨¼ÓËÙ¶Èa £½6i £«4j,ʽÖÐaµÄµ¥Î»Îªm¡¤£ó £®ÔÚt£½0ʱ,ÆäËÙ¶ÈΪÁã,λÖÃʸÁ¿r0 £½10 mi£®Çó£º(1) ÔÚÈÎÒâʱ¿ÌµÄËٶȺÍλÖÃʸÁ¿£»(2) ÖʵãÔÚOxy ƽÃæÉϵĹ켣·½³Ì,²¢»³ö¹ì¼£µÄʾÒâͼ£®
Ìâ 1-11 ͼ
·ÖÎö ÓëÉÏÁ½Ìⲻͬ´¦ÔÚÓÚÖʵã×÷ƽÃæÇúÏßÔ˶¯,¸ù¾Ýµþ¼ÓÔÀí,Çó½âʱÐè¸ù¾Ý¼ÓËٶȵÄÁ½¸ö·ÖÁ¿ax ºÍay·Ö±ð»ý·Ö,´Ó¶øµÃµ½Ô˶¯·½³ÌrµÄÁ½¸ö·ÖÁ¿Ê½x(t)ºÍy(t)£®ÓÉÓÚ±¾ÌâÖÐÖʵã¼ÓËÙ¶ÈΪºãʸÁ¿,¹ÊÁ½´Î»ý·ÖºóËùµÃÔ˶¯·½³ÌΪ¹Ì¶¨ÐÎʽ,¼´x11?x0?v0xt?axt2ºÍy?y0?v0yt?ayt2,Á½¸ö·ÖÔ˶¯¾ùΪÔȱäËÙÖ±
22ÏßÔ˶¯£®¶ÁÕß²»·Á×Ô¼ºÑé֤һϣ®
½â ÓɼÓËٶȶ¨Òåʽ,¸ù¾Ý³õʼÌõ¼þt0 £½0ʱv0 £½0,»ý·Ö¿ÉµÃ
?ÓÖÓÉvv0dv??adt??(6i?4j)dt
00ttv?6ti?4tj
?drdt¼°³õʼÌõ¼þt£½0 ʱ,r0£½(10 m)i,»ý·Ö¿ÉµÃ
?rr0dr??vdt??(6ti?4tj)dt
00ttr?(10?3t2)i?2t2j
7
ÓÉÉÏÊö½á¹û¿ÉµÃÖʵãÔ˶¯·½³ÌµÄ·ÖÁ¿Ê½,¼´
x £½10£«3t2
y £½2t2
ÏûÈ¥²ÎÊýt,¿ÉµÃÔ˶¯µÄ¹ì¼£·½³Ì
3y £½2x -20 m
ÕâÊÇÒ»¸öÖ±Ïß·½³Ì£®Ö±ÏßбÂÊk?dy2?tan¦Á?,¦Á£½33¡ã41¡ä£®¹ì¼£ÈçͼËùʾ£® dx32
1 -12 ÖʵãÔÚOxy ƽÃæÄÚÔ˶¯,ÆäÔ˶¯·½³ÌΪr£½2.0ti £«(19.0 -2.0t )j,ʽÖÐr µÄµ¥Î»Îªm,tµÄµ¥Î»Îªs£®Çó£º(1)ÖʵãµÄ¹ì¼£·½³Ì£»(2) ÔÚt1£½1.0s µ½t2 £½2.0s ʱ¼äÄÚµÄƽ¾ùËٶȣ»(3) t1 £½1.0£óʱµÄËٶȼ°ÇÐÏòºÍ·¨Ïò¼ÓËٶȣ»(4) t £½1.0s ʱÖʵãËùÔÚ´¦¹ìµÀµÄÇúÂʰ뾶¦Ñ£®
·ÖÎö ¸ù¾ÝÔ˶¯·½³Ì¿ÉÖ±½Óд³öÆä·ÖÁ¿Ê½x £½x(t)ºÍy £½y(t),´ÓÖÐÏûÈ¥²ÎÊýt,¼´µÃÖʵãµÄ¹ì¼£·½³Ì£®Æ½¾ùËÙ¶ÈÊÇ·´Ó³ÖʵãÔÚÒ»¶Îʱ¼äÄÚλÖõı仯ÂÊ,¼´v¾ùËٶȵļ«ÏÞ¼´Ë²Ê±ËÙ¶Èv?¦¤r¦¤t,ËüÓëʱ¼ä¼ä¸ô¦¤t µÄ´óСÓйØ,µ±¦¤t¡ú0 ʱ,ƽ
?drdt£®ÇÐÏòºÍ·¨Ïò¼ÓËÙ¶ÈÊÇÖ¸ÔÚ×ÔÈ»×ø±êϵķÖʸÁ¿a£ô ºÍan ,Ç°ÕßÖ»·´Ó³ÖÊ
µãÔÚÇÐÏß·½ÏòËٶȴóСµÄ±ä»¯ÂÊ,¼´at?dvte,ºóÕßÖ»·´Ó³ÖʵãËٶȷ½ÏòµÄ±ä»¯,Ëü¿ÉÓÉ×ܼÓËÙ¶Èa ºÍadtÇó¦Ñ£®
£ô
v2µÃµ½£®ÔÚÇóµÃt ʱ¿ÌÖʵãµÄËٶȺͷ¨Ïò¼ÓËٶȵĴóСºó,¿ÉÓɹ«Ê½an?¦Ñ1
½â (1) ÓɲÎÊý·½³Ì
x £½2.0t, y £½19.0-2.0t2
ÏûÈ¥t µÃÖʵãµÄ¹ì¼£·½³Ì£º
y £½19.0 -0.50x2
(2) ÔÚt1 £½1.00£ó µ½t2 £½2.0£óʱ¼äÄÚµÄƽ¾ùËÙ¶È
v?¦¤rr2?r1??2.0i?6.0j ¦¤tt2?t1(3) ÖʵãÔÚÈÎÒâʱ¿ÌµÄËٶȺͼÓËٶȷֱðΪ
v(t)?vxi?vyj?dxdyi?j?2.0i?4.0tj dtdtd2xd2ya(t)?2i?2j??4.0m?s?2j
dtdtÔòt1 £½1.00£óʱµÄËÙ¶È
v(t)£üt £½1£ó£½2.0i -4.0j ÇÐÏòºÍ·¨Ïò¼ÓËٶȷֱðΪ
att?1s?dvd2?2et?(vx?v2)e?3.58m?set ytdtdtan?a2?at2en?1.79m?s?2en
8
(4) t £½1.0£óÖʵãµÄËٶȴóСΪ
2?1v?vx?v2y?4.47m?s
v2Ôò¦Ñ??11.17m
an1 -13 ·É»úÒÔ100 m¡¤£ó µÄËÙ¶ÈÑØˮƽֱÏß·ÉÐÐ,ÔÚÀëµØÃæ¸ßΪ100 mʱ,¼ÝʻԱҪ°ÑÎïÆ·¿ÕͶµ½Ç°·½Ä³Ò»µØÃæÄ¿±ê´¦,ÎÊ£º(1) ´ËʱĿ±êÔÚ·É»úÕýÏ·½Î»ÖõÄÇ°Ãæ¶àÔ¶£¿ (2) Ͷ·ÅÎïƷʱ,¼ÝʻԱ¿´Ä¿±êµÄÊÓÏߺÍˮƽÏ߳ɺνǶȣ¿(3) ÎïƷͶ³ö2.0£óºó,ËüµÄ·¨Ïò¼ÓËٶȺÍÇÐÏò¼ÓËٶȸ÷Ϊ¶àÉÙ£¿
-1
Ìâ 1-13 ͼ
·ÖÎö ÎïÆ·¿ÕͶºó×÷ƽÅ×Ô˶¯£®ºöÂÔ¿ÕÆø×èÁ¦µÄÌõ¼þÏÂ,ÓÉÔ˶¯¶ÀÁ¢ÐÔÔÀíÖª,ÎïÆ·ÔÚ¿ÕÖÐÑØˮƽ·½Ïò×÷ÔÈËÙÖ±ÏßÔ˶¯,ÔÚÊúÖ±·½Ïò×÷×ÔÓÉÂäÌåÔ˶¯£®µ½´ïµØÃæÄ¿±êʱ,Á½·½ÏòÉÏÔ˶¯Ê±¼äÊÇÏàͬµÄ£®Òò´Ë,·Ö±ðÁгöÆäÔ˶¯·½³Ì,ÔËÓÃʱ¼äÏàµÈµÄÌõ¼þ,¼´¿ÉÇó½â£®
´ËÍâ,ƽÅ×ÎïÌåÔÚÔ˶¯¹ý³ÌÖÐÖ»´æÔÚÊúÖ±ÏòϵÄÖØÁ¦¼ÓËٶȣ®ÎªÇóÌض¨Ê±¿ÌtʱÎïÌåµÄÇÐÏò¼ÓËٶȺͷ¨Ïò¼ÓËÙ¶È,Ö»ÐèÇó³ö¸Ãʱ¿ÌËüÃÇÓëÖØÁ¦¼ÓËÙ¶ÈÖ®¼äµÄ¼Ð½Ç¦Á»ò¦Â£®ÓÉͼ¿ÉÖª,ÔÚÌض¨Ê±¿Ìt,ÎïÌåµÄÇÐÏò¼ÓËٶȺÍˮƽÏßÖ®¼äµÄ¼Ð½Ç¦Á,¿ÉÓÉ´Ëʱ¿ÌµÄÁ½ËٶȷÖÁ¿vx ¡¢vyÇó³ö,ÕâÑù,Ò²¾Í¿É½«ÖØÁ¦¼ÓËÙ¶Èg µÄÇÐÏòºÍ·¨Ïò·ÖÁ¿ÇóµÃ£®
½â (1) È¡ÈçͼËùʾµÄ×ø±ê,ÎïÆ·ÏÂÂäʱÔÚˮƽºÍÊúÖ±·½ÏòµÄÔ˶¯·½³Ì·Ö±ðΪ
x £½vt, y £½1/2 gt2
·É»úˮƽ·ÉÐÐËÙ¶Èv£½100 m¡¤s ,·É»úÀëµØÃæµÄ¸ß¶Èy£½100 m,ÓÉÉÏÊöÁ½Ê½¿ÉµÃÄ¿±êÔÚ·É»úÕýÏ·½Ç°µÄ¾àÀë
-1
x?v(2) ÊÓÏߺÍˮƽÏߵļнÇΪ
2y?452m gy?12.5o x¦È?arctan(3) ÔÚÈÎÒâʱ¿ÌÎïÆ·µÄËÙ¶ÈÓëˮƽÖáµÄ¼Ð½ÇΪ
¦Á?arctanvyvx?arctangtv
È¡×ÔÈ»×ø±ê,ÎïÆ·ÔÚÅ׳ö2s ʱ,ÖØÁ¦¼ÓËٶȵÄÇÐÏò·ÖÁ¿Óë·¨Ïò·ÖÁ¿·Ö±ðΪ
gt??at?gsin¦Á?gsin?arctan??1.88m?s?2
v??
9
gt??an?gcos??gcos?arctan??9.62m?s?2
v??1 -14 ΪӽÓÏã¸Û»Ø¹é£¬Ìؼ¼ÑÝÔ±¿ÂÊÜÁ¼ÔÚ1997Äê6ÔÂ1Èռݳµ·ÉÔ½»ÆºÓºø¿Ú£¬ÈçͼËùʾ£¬¿Â¼Ý³µ´ÓÅܵÀ¶«¶ËÆô¶¯£¬µ½´ïÅܵÀÖÕ¶ËʱËٶȴóСΪ
v0?150 km?h?1 £¬ËûËæ¼´ÒÔÑö½Ç??5?³å³ö£¬·ÉÔ½¿ç¶È´ï57 m£¬°²È«×ŽÔÚÎ÷°¶Ä¾ÇÅÉÏ£¬Çó£º
Ìâ 1-14 ͼ
£¨1£© ¿Â·É³µ¿çÔ½»ÆºÓÓÃÁ˶೤ʱ¼ä£¿
£¨2£© ÈôÆð·Éµã¸ß³öºÓÃæ10 m£¬¿Â¼Ý³µ·ÉÐеÄ×î¸ßµã¾àºÓÃæΪ¼¸Ã×£¿ £¨3£© Î÷°¶Ä¾ÇźÍÆð·ÉµãµÄ¸ß¶È²îΪ¶àÉÙ£¿
·ÖÎö ÓÉÌâÒâÖª£¬·É³µ×÷бÉÏÅ×Ô˶¯£¬¶Ô°üº¬Å×ÌåÔÚÄÚµÄÒ»°ãÇúÏßÔ˶¯
À´Ëµ£¬ÔËÓõþ¼ÓÔÀíÊÇÇó½â´ËÀàÎÊÌâµÄÆÕÊÊ·½·¨£¬²Ù×÷³ÌÐòÊÇ£º½¨Á¢Ò»¸öÇ¡µ±µÄÖ±½Ç×ø±êϵ£¬½«Ô˶¯·Ö½âΪÁ½¸öÏ໥Õý½»µÄÖ±ÏßÔ˶¯£¬ÓÉÓÚÔÚÅ×ÌåÔ˶¯ÖУ¬ÖʵãµÄ¼ÓËٶȺãΪg£¬¹ÊÁ½¸ö·ÖÔ˶¯¾ùΪÔȱäËÙÖ±ÏßÔ˶¯»òÆäÖÐÒ»¸öΪÔÈËÙÖ±ÏßÔ˶¯£¬Ö±½ÓÁгöÏà¹ØÔ˶¯¹æÂÉ·½³Ì¼´¿ÉÇó½â£¬±¾Ìâ¿É½¨Á¢Í¼Ê¾×ø±êϵ£¬Í¼Öзֱð±íʾ·É³µµÄ×î´ó¸ß¶ÈºÍ·ÉÔ¾¿ç¶È. ½â ÔÚͼʾ×ø±êϵÖУ¬ÓÐ
x
ymºÍxm?(v0cos?)t £¨1£©
y?(v0sin?£©t?12gt2 £¨2£©
vy(1) ÓÉʽ£¨1£©£¬Áîx?v0sin??gt £¨3£©
?xm?57 m£¬µÃ·ÉԾʱ¼ä
tm?xm?1.37 s
v0cos?£¨2£©ÓÉʽ£¨3£©£¬Áîvy?0£¬µÃ·ÉÐе½×î´ó¸ß¶ÈËùÐèʱ¼ä
t¡¯m?½«tm´úÈëʽ£¨2£©£¬µÃ·ÉÐÐ×î´ó¸ß¶È
¡¯v0sin?g
10
2v0sin2?ym??0.67m
2gÔò·É³µÔÚ×î¸ßµãʱ¾àºÓÃæ¾àÀëΪ
h?ym?10 m?10.67 m
£¨3£©½«tm?1.37 s ´úÈëʽ£¨2£©£¬µÃÎ÷°¶Ä¾ÇÅλÖÃΪ
y = - 4.22 m
¡°-¡±ºÅ±íʾľÇÅÔڷɳµÆð·ÉµãµÄÏ·½.
ÌÖÂÛ ±¾ÌâÒ²¿ÉÒÔË®ÃæΪ×ø±êϵԵ㣬Ôò·É³µÔÚ y·½ÏòÉϵÄÔ˶¯·½³ÌӦΪ
y?10 m + (v0sin?)t?12gt 2?30?£¬ÇòµÄÅ×Éä½Ç??60?£¬Éè
1 -15 ÈçͼËùʾ£¬´ÓɽƵ׶˽«Ð¡ÇòÅ׳ö£¬ÒÑÖª¸ÃɽÆÂÓк㶨Çã½Ç?-1
Çò±»Å׳öʱµÄËÙÂÊv0 £½19.6 m¡¤£ó£¬ºöÂÔ¿ÕÆø×èÁ¦£¬ÎÊÇòÂäÔÚɽÆÂÉÏ´¦ÀëɽƵ׶˵ľàÀëΪ¶àÉÙ£¿´Ë¹ý
³Ì¾Àú¶à³¤Ê±¼ä£¿
Ìâ 1-15 ͼ
·ÖÎö Çó½â·½·¨ÓëÉÏÌâÀàËÆ£¬µ«±¾Ìâ¿É½«Ô˶¯°´Á½ÖÖ·½Ê½·Ö½â£¬Èçͼ£¨a£©ºÍͼ£¨b£©Ëùʾ.ÔÚͼ£¨a£©×ø±êϵÖУ¬Á½¸ö·ÖÔ˶¯¾ùΪÔȼõËÙÖ±ÏßÔ˶¯£¬¼ÓËٶȴóС·Ö±ðΪ-g cos?ºÍ-g sin?£¬¿´ËƸ´ÔÓ£¬µ«Çó½â±¾ÌâÈ·½Ï·½±ã£¬ÒòΪÂäµØʱÓÐy=0,¶ÔÓ¦µÄʱ¼ätºÍxµÄÖµ¼´Îª±¾ÌâËùÇó.ÔÚͼ£¨b£©×ø±êϵÖУ¬·ÖÔ˶¯¿´ËƼòµ¥£¬µ«Çó½â±¾Ì⻹Ð轫ÂäµØµãPµÄ×ø±êyÓëxµÄ¹ØϵÁгöÀ´. ½â 1 ÓÉ·ÖÎöÖª£¬ÔÚͼ£¨a£©×ø±êϵÖУ¬ÓÐ
x1?[v0cos(???)]t?(?gsin?)t2 £¨1£©
212 y?[v0sin(???)]t?(?gcos?)t £¨2£©
2
ÂäµØʱ£¬ÓÐy=0£¬ÓÉʽ£¨2£©½âµÃ·ÉÐÐʱ¼äΪ
t?½« t Öµ´úÈëʽ£¨1£©£¬µÃ
2v0tan30??2.31s g 11
22v0OP?x??26.1m
3g
½â 2 ÓÉ·ÖÎöÖª£¬ÔÚͼ£¨b£©×ø±êϵÖУ¬ ¶ÔСÇò x?(v0cos?)t £¨1£©
¶ÔµãP
y?(v0sin?)t?12gt £¨2£© 2y??xtan? £¨3£©
ÓÉʽ£¨1£©¡¢£¨2£©¿ÉµÃÇòµÄ¹ìµÀ·½³ÌΪ
gx2 y?xtan??22v0cos2??ÂäµØʱ£¬Ó¦ÓÐ
£¨4£©
y?y?£¬¼´
gx2xtan30?xtan60?22v0cos260???
½âÖ®µÃÂäµØµãPµÄx×ø±êΪ
23v0 x?3g £¨5£©
22v0x??26.1m Ôò OP?cos30?3gÁª½âʽ£¨1£©ºÍʽ£¨5£©¿ÉµÃ·ÉÐÐʱ¼ä
t?2.31s
ÌÖÂÛ ±È½ÏÁ½Öֽⷨ£¬Äã¶ÔÈçºÎÁé»îÔËÓõþ¼ÓÔÀíÓÐʲôÌå»á?
1 -16 Ò»ÖʵãÑذ뾶ΪR µÄÔ²ÖÜ°´¹æÂÉs1?v0t?bt2Ô˶¯,v ¡¢b ¶¼Êdz£Á¿£®(1) Çót ʱ¿ÌÖʵãµÄ×Ü
20
¼ÓËٶȣ»(2) t ΪºÎֵʱ×ܼÓËÙ¶ÈÔÚÊýÖµÉϵÈÓÚb£¿(3) µ±¼ÓËٶȴﵽb ʱ,ÖʵãÒÑÑØÔ²ÖÜÔËÐÐÁ˶àÉÙȦ£¿ ·ÖÎö ÔÚ×ÔÈ»×ø±êÖÐ,s ±íʾԲÖÜÉÏ´Óijһµã¿ªÊ¼µÄÇúÏß×ø±ê£®Óɸø¶¨µÄÔ˶¯·½³Ìs £½s(t),¶Ôʱ¼ät ÇóÒ»½×¡¢¶þ½×µ¼Êý,¼´ÊÇÑØÇúÏßÔ˶¯µÄËÙ¶Èv ºÍ¼ÓËٶȵÄÇÐÏò·ÖÁ¿a£ô,¶ø¼ÓËٶȵķ¨Ïò·ÖÁ¿Îªan£½v2 /R£®ÕâÑù,×ܼÓËÙ¶ÈΪa £½a£ôe£ô£«anen£®ÖÁÓÚÖʵãÔÚt ʱ¼äÄÚͨ¹ýµÄ·³Ì,¼´ÎªÇúÏß×ø±êµÄ¸Ä±äÁ¿¦¤s£½st -s0£®ÒòÔ²Öܳ¤Îª2¦ÐR,ÖʵãËùת¹ýµÄȦÊý×ÔÈ»¿ÉÇóµÃ£® ½â (1) Öʵã×÷Ô²ÖÜÔ˶¯µÄËÙÂÊΪ
v?Æä¼ÓËٶȵÄÇÐÏò·ÖÁ¿ºÍ·¨Ïò·ÖÁ¿·Ö±ðΪ
ds?v0?bt dt 12
d2sv2(v0?bt)2at?2??b, an??dtRR¹Ê¼ÓËٶȵĴóСΪ
at2b2?(v0?bt)4a?a?a?R2n2t
Æä·½ÏòÓëÇÐÏßÖ®¼äµÄ¼Ð½ÇΪ
?(v0?bt)2?an¦È?arctan?arctan?? ?atRb??(2) Ҫʹ£üa£ü£½b,ÓÉ
1R2b2?(v0?bt)4?b¿ÉµÃ Rvt?0
b2v0s?st?s0?
2b(3) ´Ót£½0 ¿ªÊ¼µ½t£½v0 /b ʱ,Öʵ㾹ýµÄ·³ÌΪ
Òò´ËÖʵãÔËÐеÄȦÊýΪ
2sv0n??
2¦ÐR4¦ÐbR1 -17 Ò»°ë¾¶Îª0.50 m µÄ·ÉÂÖÔÚÆô¶¯Ê±µÄ¶Ìʱ¼äÄÚ,Æä½ÇËÙ¶ÈÓëʱ¼äµÄƽ·½³ÉÕý±È£®ÔÚt£½2.0£ó ʱ²âµÃÂÖÔµÒ»µãµÄËÙ¶ÈֵΪ4.0 m¡¤£ó£®Çó£º(1) ¸ÃÂÖÔÚt¡ä£½0.5£óµÄ½ÇËÙ¶È,ÂÖÔµÒ»µãµÄÇÐÏò¼ÓËٶȺÍ×ܼÓËٶȣ»(2)¸ÃµãÔÚ2.0£óÄÚËùת¹ýµÄ½Ç¶È£®
·ÖÎö Ê×ÏÈÓ¦¸ÃÈ·¶¨½ÇËٶȵĺ¯Êý¹Øϵ¦Ø£½kt£®ÒÀ¾Ý½ÇÁ¿ÓëÏßÁ¿µÄ¹ØϵÓÉÌض¨Ê±¿ÌµÄËÙ¶ÈÖµ¿ÉµÃÏàÓ¦µÄ½ÇËÙ¶È,´Ó¶øÇó³öʽÖеıÈÀýϵÊýk,¦Ø£½¦Ø(t)È·¶¨ºó,×¢Òâµ½Ô˶¯µÄ½ÇÁ¿ÃèÊöÓëÏßÁ¿ÃèÊöµÄÏàÓ¦¹Øϵ,ÓÉÔ˶¯Ñ§ÖÐÁ½ÀàÎÊÌâÇó½âµÄ·½·¨(΢·Ö·¨ºÍ»ý·Ö·¨),¼´¿ÉµÃµ½Ìض¨Ê±¿ÌµÄ½Ç¼ÓËٶȡ¢ÇÐÏò¼ÓËٶȺͽÇλÒÆ£® ½â Òò¦ØR £½v,ÓÉÌâÒâ¦Ø¡Øt µÃ±ÈÀýϵÊý
2
2
-1
k?ËùÒÔ ¦Ø¦Øv?3 ??2rad?s22tRt?¦Ø(t)?2t2
¦Ø?2t?2?0.5rad?s?1
Ôòt¡ä£½0.5£ó ʱµÄ½ÇËٶȡ¢½Ç¼ÓËٶȺÍÇÐÏò¼ÓËٶȷֱðΪ
¦Á?d¦Ø?4t??2.0rad?s?2 dtat?¦ÁR?1.0m?s?2
×ܼÓËÙ¶È
a?an?at?¦ÁRet?¦Ø2Ren
13
a?ÔÚ2.0£óÄڸõãËùת¹ýµÄ½Ç¶È
2?¦ÁR?2??¦Ø2R?22?1.01m?s?2
1 -18
232t0?5.33rad
0033Ò»ÖʵãÔڰ뾶Ϊ0.10 mµÄÔ²ÖÜÉÏÔ˶¯,Æä½ÇλÖÃΪ¦È?2?4t,ʽÖÐ¦È µÄµ¥Î»Îªrad,t µÄµ¥Î»Îª
¦È?¦È0??¦Ødt??2t2dt?£ó£®(1) ÇóÔÚt £½2.0£óʱÖʵãµÄ·¨Ïò¼ÓËٶȺÍÇÐÏò¼ÓËٶȣ®(2) µ±ÇÐÏò¼ÓËٶȵĴóСǡµÈÓÚ×ܼÓËٶȴóСµÄÒ»°ëʱ,¦È ֵΪ¶àÉÙ£¿(3) t Ϊ¶àÉÙʱ,·¨Ïò¼ÓËٶȺÍÇÐÏò¼ÓËٶȵÄÖµÏàµÈ£¿
·ÖÎö ÕÆÎÕ½ÇÁ¿ÓëÏßÁ¿¡¢½ÇλÒÆ·½³ÌÓëλʸ·½³ÌµÄ¶ÔÓ¦¹Øϵ,Ó¦ÓÃÔ˶¯Ñ§Çó½âµÄ·½·¨¼´¿ÉµÃµ½£® ½â (1) ÓÉÓڦȷֱðΪ
?2?4t3,Ôò½ÇËٶȦØ?d¦È?12t2£®ÔÚt £½2 £ó ʱ,·¨Ïò¼ÓËٶȺÍÇÐÏò¼ÓËٶȵÄÊýÖµdtanat(2) µ±att?2s?r¦Ø2?2.30m?s?2 ?r2t?2sd¦Ø?4.80m?s?2 dt?a/2?12an?at22ʱ,ÓÐ3at2?an,¼´
3?24rt??r212t22??
4µÃ t3?123
´Ëʱ¿ÌµÄ½ÇλÖÃΪ
¦È?2?4t3?3.15rad
(3) Ҫʹan?at,ÔòÓÐ
23?24rt??r212t2t £½0.55£ó
-1
??
41 -19 Ò»ÎÞ·çµÄÏÂÓêÌì,Ò»Áлð³µÒÔv1£½20.0 m¡¤£ó µÄËÙ¶ÈÔÈËÙÇ°½ø,ÔÚ³µÄÚµÄÂÿͿ´¼û²£Á§´°ÍâµÄÓêµÎºÍ´¹Ïß³É75¡ã½ÇϽµ£®ÇóÓêµÎÏÂÂäµÄËÙ¶Èv2 £®(ÉèϽµµÄÓêµÎ×÷ÔÈËÙÔ˶¯)
Ìâ 1-19 ͼ
·ÖÎö ÕâÊÇÒ»¸öÏà¶ÔÔ˶¯µÄÎÊÌ⣮ÉèÓêµÎΪÑо¿¶ÔÏó,µØÃæΪ¾²Ö¹²Î¿¼Ïµ£Ó,»ð³µÎª¶¯²Î¿¼Ïµ£Ó¡ä£®v1 Ϊ£Ó¡äÏà¶Ô£Ó µÄËÙ¶È,v2 ΪÓêµÎÏà¶Ô£ÓµÄËÙ¶È,ÀûÓÃÏà¶ÔÔ˶¯ËٶȵĹØϵ¼´¿É½â£®½â ÒÔµØÃæΪ²Î¿¼Ïµ,»ð³µÏà¶ÔµØÃæÔ˶¯µÄËÙ¶ÈΪv1 ,ÓêµÎÏà¶ÔµØÃæÊúÖ±ÏÂÂäµÄËÙ¶ÈΪv2 ,ÂÿͿ´µ½ÓêµÎÏÂÂäµÄËÙ¶Èv2¡äΪÏà¶ÔËÙ¶È,ËüÃÇÖ®¼äµÄ¹ØϵΪv2?v2?v1 (ÈçͼËùʾ),ÓÚÊǿɵÃ
'v2?v1?5.36m?s?1 otan7514
1 -20 Èçͼ(a)Ëùʾ,Ò»Æû³µÔÚÓêÖÐÑØÖ±ÏßÐÐÊ»,ÆäËÙÂÊΪv1 ,ÏÂÂäÓêµÎµÄËٶȷ½ÏòÆ«ÓÚÊúÖ±·½Ïò֮ǰ¦È ½Ç,ËÙÂÊΪv2¡ä,Èô³µºóÓÐÒ»³¤·½ÐÎÎïÌå,ÎʳµËÙv1Ϊ¶à´óʱ,´ËÎïÌåÕýºÃ²»»á±»ÓêË®ÁÜʪ£¿
·ÖÎö ÕâÒ²ÊÇÒ»¸öÏà¶ÔÔ˶¯µÄÎÊÌ⣮¿ÉÊÓÓêµãΪÑо¿¶ÔÏó,µØÃæΪ¾²²Î¿¼Ïµ£Ó,Æû³µÎª¶¯²Î¿¼Ïµ£Ó¡ä£®Èçͼ(a)Ëùʾ,ҪʹÎïÌå²»±»ÁÜʪ,ÔÚ³µÉϹ۲ìÓêµãÏÂÂäµÄ·½Ïò(¼´ÓêµãÏà¶ÔÓÚÆû³µµÄÔ˶¯ËÙ¶Èv2¡äµÄ·½Ïò)Ó¦Âú×ã¦Á?arctanl£®ÔÙÓÉÏà¶ÔËٶȵÄʸÁ¿¹Øϵv?2?v2?v1,¼´¿ÉÇó³öËùÐè³µËÙv£® h1
Ìâ 1-20 ͼ
½â ÓÉv?2?v2?v1£Ûͼ(b)£Ý,ÓÐ
??arctan¶øҪʹ¦Áv1?v2sin¦Èv2cos¦È
?arctanl,Ôò hv1?v2sin¦Èl?
v2cos¦Èh?lcos¦È?v1?v2??sin¦È?
?h?µÚ¶þÕ ţ¶Ù¶¨ÂÉ
2 -1 Èçͼ(a)Ëùʾ,ÖÊÁ¿Îªm µÄÎïÌåÓÃƽÐÐÓÚбÃæµÄϸÏßÁª½áÖÃÓڹ⻬µÄбÃæÉÏ,ÈôбÃæÏò×ó·½×÷¼ÓËÙÔ˶¯,µ±ÎïÌå¸ÕÍÑÀëбÃæʱ,ËüµÄ¼ÓËٶȵĴóСΪ( )
(A) gsin ¦È (B) gcos ¦È (C) gtan ¦È (D) gcot ¦È 15
·ÖÎöÓë½â µ±ÎïÌåÀ뿪бÃæ˲¼ä,бÃæ¶ÔÎïÌåµÄÖ§³ÖÁ¦ÏûʧΪÁã,ÎïÌåÔÚÉþ×ÓÀÁ¦F£Ô (Æä·½ÏòÈÔ¿ÉÈÏΪƽÐÐÓÚбÃæ)ºÍÖØÁ¦×÷ÓÃϲúÉúƽÐÐˮƽÃæÏò×óµÄ¼ÓËÙ¶Èa,Èçͼ(b)Ëùʾ,ÓÉÆä¿É½âµÃºÏÍâÁ¦Îªmgcot ¦È,¹ÊÑ¡(D)£®Çó½âµÄ¹Ø¼üÊÇÕýÈ··ÖÎöÎïÌå¸ÕÀ뿪бÃæ˲¼äµÄÎïÌåÊÜÁ¦Çé¿öºÍ״̬ÌØÕ÷£®
2 -2 ÓÃˮƽÁ¦FN°ÑÒ»¸öÎïÌåѹ×Å¿¿ÔÚ´Ö²ÚµÄÊúֱǽÃæÉϱ£³Ö¾²Ö¹£®µ±FNÖð½¥Ôö´óʱ,ÎïÌåËùÊܵľ²Ä¦²ÁÁ¦
FfµÄ´óС( )
(A) ²»ÎªÁã,µ«±£³Ö²»±ä (B) ËæFN³ÉÕý±ÈµØÔö´ó
(C) ¿ªÊ¼ËæFNÔö´ó,´ïµ½Ä³Ò»×î´óÖµºó,¾Í±£³Ö²»±ä (D) ÎÞ·¨È·¶¨
·ÖÎöÓë½â Ó뻬¶¯Ä¦²ÁÁ¦²»Í¬µÄÊÇ,¾²Ä¦²ÁÁ¦¿ÉÔÚÁãÓë×î´óÖµ¦ÌFN·¶Î§ÄÚÈ¡Öµ£®µ±FNÔö¼Óʱ,¾²Ä¦²ÁÁ¦¿ÉÈ¡µÄ×î´óÖµ³ÉÕý±ÈÔö¼Ó,µ«¾ßÌå´óСÔòÈ¡¾öÓÚ±»×÷ÓÃÎïÌåµÄÔ˶¯×´Ì¬£®ÓÉÌâÒâÖª,ÎïÌåÒ»Ö±±£³Ö¾²Ö¹×´Ì¬,¹Ê¾²Ä¦²ÁÁ¦ÓëÖØÁ¦´óСÏàµÈ,·½ÏòÏà·´,²¢±£³Ö²»±ä,¹ÊÑ¡(A)£®
2 -3 Ò»¶Î·ÃæˮƽµÄ¹«Â·,תÍä´¦¹ìµÀ°ë¾¶ÎªR,Æû³µÂÖÌ¥Óë·Ãæ¼äµÄĦ²ÁÒòÊýΪ¦Ì,ҪʹÆû³µ²»ÖÁÓÚ·¢Éú²àÏò´ò»¬,Æû³µÔڸô¦µÄÐÐÊ»ËÙÂÊ( ) (A) ²»µÃСÓÚ(C) ²»µÃ´óÓÚ
¦ÌgR (B) ±ØÐëµÈÓÚ¦ÌgR
¦ÌgR (D) »¹Ó¦ÓÉÆû³µµÄÖÊÁ¿m ¾ö¶¨
·ÖÎöÓë½â ÓÉÌâÒâÖª,Æû³µÓ¦ÔÚˮƽÃæÄÚ×÷ÔÈËÙÂÊÔ²ÖÜÔ˶¯,Ϊ±£Ö¤Æû³µ×ªÍäʱ²»²àÏò´ò»¬,ËùÐèÏòÐÄÁ¦Ö»ÄÜÓÉ·ÃæÓëÂÖÌ¥¼äµÄ¾²Ä¦²ÁÁ¦Ìṩ,Äܹ»ÌṩµÄ×î´óÏòÐÄÁ¦Ó¦Îª¦ÌFN£®ÓÉ´Ë¿ÉËãµÃÆû³µ×ªÍäµÄ×î´óËÙÂÊӦΪ
v£½¦ÌRg£®Òò´ËÖ»ÒªÆû³µ×ªÍäʱµÄʵ¼ÊËÙÂʲ»´óÓÚ´ËÖµ,¾ùÄܱ£Ö¤²»²àÏò´ò»¬£®Ó¦Ñ¡(C)£®
2 -4 Ò»ÎïÌåÑع̶¨Ô²»¡Ðι⻬¹ìµÀÓɾ²Ö¹Ï»¬,ÔÚÏ»¬¹ý³ÌÖÐ,Ôò( ) (A) ËüµÄ¼ÓËٶȷ½ÏòÓÀÔ¶Ö¸ÏòÔ²ÐÄ,ÆäËÙÂʱ£³Ö²»±ä (B) ËüÊܵ½µÄ¹ìµÀµÄ×÷ÓÃÁ¦µÄ´óС²»¶ÏÔö¼Ó (C) ËüÊܵ½µÄºÏÍâÁ¦´óС±ä»¯,·½ÏòÓÀÔ¶Ö¸ÏòÔ²ÐÄ (D) ËüÊܵ½µÄºÏÍâÁ¦´óС²»±ä,ÆäËÙÂʲ»¶ÏÔö¼Ó
16
·ÖÎöÓë½â ÓÉͼ¿ÉÖª,ÎïÌåÔÚÏ»¬¹ý³ÌÖÐÊܵ½´óСºÍ·½Ïò²»±äµÄÖØÁ¦ÒÔ¼°Ê±¿ÌÖ¸ÏòÔ²¹ìµÀÖÐÐĵĹìµÀÖ§³ÖÁ¦FN×÷ÓÃ,ÆäºÏÍâÁ¦·½Ïò²¢·ÇÖ¸ÏòÔ²ÐÄ,Æä´óСºÍ·½Ïò¾ùÓëÎïÌåËùÔÚλÖÃÓйأ®ÖØÁ¦µÄÇÐÏò·ÖÁ¿(m gcos ¦È) ʹÎïÌåµÄËÙÂʽ«»á²»¶ÏÔö¼Ó(ÓÉ»úеÄÜÊغãÒà¿ÉÅжÏ),ÔòÎïÌå×÷Ô²ÖÜÔ˶¯µÄÏòÐÄÁ¦(ÓֳƷ¨ÏòÁ¦)½«²»¶ÏÔö
v2´ó,ÓɹìµÀ·¨Ïò·½ÏòÉϵĶ¯Á¦Ñ§·½³ÌFN?mgsin¦È?mRÁ¦FNÒ²½«²»¶ÏÔö´ó,Óɴ˿ɼûӦѡ(B)£®
¿ÉÅжÏ,Ëæ¦È ½ÇµÄ²»¶ÏÔö´ó¹ý³Ì,¹ìµÀÖ§³Ö
*2 -5 ͼ(a)ʾϵͳÖÃÓÚÒÔa £½1/4 g µÄ¼ÓËÙ¶ÈÉÏÉýµÄÉý½µ»úÄÚ,A¡¢B Á½ÎïÌåÖÊÁ¿Ïàͬ¾ùΪm,A ËùÔÚµÄ×ÀÃæÊÇˮƽµÄ,Éþ×ӺͶ¨»¬ÂÖÖÊÁ¿¾ù²»¼Æ,ÈôºöÂÔ»¬ÂÖÖáÉϺÍ×ÀÃæÉϵÄĦ²Á,²¢²»¼Æ¿ÕÆø×èÁ¦,ÔòÉþÖÐÕÅÁ¦Îª( )
(A) 5/8 mg (B) 1/2 mg (C) mg (D) 2mg
·ÖÎöÓë½â ±¾Ìâ¿É¿¼ÂǶÔA¡¢B Á½ÎïÌå¼ÓÉϹßÐÔÁ¦ºó,ÒÔµçÌÝÕâ¸ö·Ç¹ßÐԲο¼Ïµ½øÐÐÇó½â£®´ËʱA¡¢B Á½ÎïÌåÊÜÁ¦Çé¿öÈçͼ(b)Ëùʾ,ͼÖÐa¡äΪA¡¢B Á½ÎïÌåÏà¶ÔµçÌݵļÓËÙ¶È,maΪ¹ßÐÔÁ¦£®¶ÔA¡¢B Á½ÎïÌåÓ¦ÓÃÅ£¶ÙµÚ¶þ¶¨ÂÉ,¿É½âµÃF£Ô £½5/8 mg£®¹ÊÑ¡(A)£®
ÌÖÂÛ ¶ÔÓÚÏ°Ìâ2 -5 ÕâÖÖÀàÐ͵ÄÎïÀíÎÊÌâ,ÍùÍù´Ó·Ç¹ßÐԲο¼Ïµ(±¾ÌâΪµçÌÝ)¹Û²ìµ½µÄÔ˶¯Í¼Ïñ½ÏΪÃ÷È·,µ«ÓÉÓÚÅ£¶Ù¶¨ÂÉÖ»ÊÊÓÃÓÚ¹ßÐԲο¼Ïµ,¹Ê´Ó·Ç¹ßÐԲο¼ÏµÇó½âÁ¦Ñ§ÎÊÌâʱ,±ØÐë¶ÔÎïÌå¼ÓÉÏÒ»¸öÐéÄâµÄ¹ßÐÔÁ¦£®ÈçÒÔµØÃæΪ¹ßÐԲο¼ÏµÇó½â,ÔòÁ½ÎïÌåµÄ¼ÓËÙ¶ÈaA ºÍaB ¾ùÓ¦¶ÔµØ¶øÑÔ,±¾ÌâÖÐaA ºÍaBµÄ´óСÓë·½Ïò¾ù²»Ïàͬ£®ÆäÖÐaA ӦбÏòÉÏ£®¶ÔaA ¡¢aB ¡¢a ºÍa¡äÖ®¼ä»¹ÒªÓõ½Ïà¶ÔÔ˶¯¹æÂÉ,Çó½â¹ý³Ì½Ï·±Ëö£®ÓÐÐËȤµÄ¶ÁÕß²»·Á×Ô¼º³¢ÊÔһϣ®
2 -6 ͼʾһбÃæ,Çã½ÇΪ¦Á,µ×±ßAB ³¤Îªl £½2.1 m,ÖÊÁ¿Îªm µÄÎïÌå´ÓÌâ2 -6 ͼбÃ涥¶ËÓɾ²Ö¹¿ªÊ¼ÏòÏ»¬¶¯,бÃæµÄĦ²ÁÒòÊýΪ¦Ì£½0.14£®ÊÔÎÊ,µ±¦ÁΪºÎֵʱ,ÎïÌåÔÚбÃæÉÏÏ»¬µÄʱ¼ä×î¶Ì£¿ ÆäÊýֵΪ¶àÉÙ£¿
17
·ÖÎö ¶¯Á¦Ñ§ÎÊÌâÒ»°ã·ÖΪÁ½Àࣺ(1) ÒÑÖªÎïÌåÊÜÁ¦ÇóÆäÔ˶¯Çé¿ö£»(2) ÒÑÖªÎïÌåµÄÔ˶¯Çé¿öÀ´·ÖÎöÆäËùÊܵÄÁ¦£®µ±È»,ÔÚÒ»¸ö¾ßÌåÌâÄ¿ÖÐ,ÕâÁ½ÀàÎÊÌâ²¢ÎÞ½ØÈ»µÄ½çÏÞ,ÇÒ¶¼ÊÇÒÔ¼ÓËÙ¶È×÷ΪÖнé,°Ñ¶¯Á¦Ñ§·½³ÌºÍÔ˶¯Ñ§¹æÂÉÁªÏµÆðÀ´£®±¾Ìâ¹Ø¼üÔÚÁгö¶¯Á¦Ñ§ºÍÔ˶¯Ñ§·½³Ìºó,½â³öÇã½ÇÓëʱ¼äµÄº¯Êý¹Øϵ¦Á£½f(t),È»ºóÔËÓöÔt Çó¼«ÖµµÄ·½·¨¼´¿ÉµÃ³öÊýÖµÀ´£®
½â È¡ÑØбÃæΪ×ø±êÖáOx,ÔµãO λÓÚбÃ涥µã,ÔòÓÉÅ£¶ÙµÚ¶þ¶¨ÂÉÓÐmgsin¦Á?mg¦Ìcos¦Á?ma
(1)
ÓÖÎïÌåÔÚбÃæÉÏ×÷ÔȱäËÙÖ±ÏßÔ˶¯,¹ÊÓÐ
l11?at2?g?sin¦Á?¦Ìcos¦Á?t2
cos¦Á22Ôò t?2l (2)
gcos¦Á?sin¦Á?¦Ìcos¦Á?dt?0,ÓÉʽ(2)ÓÐ d¦Á?sin¦Á?sin¦Á?¦Ìcos¦Á??cos¦Á?cos¦Á?¦Ìsin¦Á??0
??1o,??49 ¦ÌΪʹÏ»¬µÄʱ¼ä×î¶Ì,¿ÉÁî
Ôò¿ÉµÃ tan2¦Á´Ëʱ tmin?2l?0.99s
gcos??sin???cos??2
2 -7 ¹¤µØÉÏÓÐÒ»µõ³µ,½«¼×¡¢ÒÒÁ½¿é»ìÄýÍÁÔ¤ÖÆ°åµõÆðËÍÖÁ¸ß¿Õ£®¼×¿éÖÊÁ¿Îªm1 £½2.00 ¡Á10kg,ÒÒ¿éÖÊÁ¿Îªm2 £½1.00 ¡Á10kg£®Éèµõ³µ¡¢¿ò¼ÜºÍ¸ÖË¿ÉþµÄÖÊÁ¿²»¼Æ£®ÊÔÇóÏÂÊöÁ½ÖÖÇé¿öÏÂ,¸ÖË¿ÉþËùÊܵÄÕÅÁ¦ÒÔ¼°ÒÒ¿é¶Ô¼×¿éµÄ×÷ÓÃÁ¦£º(1) Á½Îï¿éÒÔ10.0 m¡¤£ó µÄ¼ÓËÙ¶ÈÉÏÉý£»(2) Á½Îï¿éÒÔ1.0 m¡¤£ó µÄ¼ÓËÙ¶ÈÉÏÉý£®´Ó±¾ÌâµÄ½á¹û,ÄãÄÜÌå»áµ½ÆðµõÖØÎïʱ±ØÐ뻺Âý¼ÓËٵĵÀÀíÂð£¿
-2
-2
2
18
Ìâ 2-7 ͼ
·ÖÎö Ô¤ÖÆ°å¡¢µõ³µ¿ò¼Ü¡¢¸ÖË¿µÈ¿ÉÊÓΪһ×éÎïÌ壮´¦Àí¶¯Á¦Ñ§ÎÊÌâͨ³£²ÉÓá°¸ôÀëÌ塱µÄ·½·¨,·ÖÎöÎïÌåËùÊܵĸ÷ÖÖ×÷ÓÃÁ¦,ÔÚËùÑ¡¶¨µÄ¹ßÐÔϵÖÐÁгöËüÃǸ÷×ԵĶ¯Á¦Ñ§·½³Ì£®¸ù¾ÝÁ¬½ÓÌåÖÐÎïÌåµÄ¶àÉÙ¿ÉÁгöÏàÓ¦ÊýÄ¿µÄ·½³Ìʽ£®½áºÏ¸÷ÎïÌåÖ®¼äµÄÏ໥×÷ÓúÍÁªÏµ,¿É½â¾öÎïÌåµÄÔ˶¯»òÏ໥×÷ÓÃÁ¦£®
½â °´ÌâÒâ,¿É·Ö±ðÈ¡µõ³µ(º¬¼×¡¢ÒÒ)ºÍÒÒ×÷Ϊ¸ôÀëÌå,»Ê¾Á¦Í¼,²¢È¡ÊúÖ±ÏòÉÏΪOy ÖáÕý·½Ïò(ÈçͼËùʾ)£®µ±¿ò¼ÜÒÔ¼ÓËÙ¶Èa ÉÏÉýʱ,ÓÐ
F£Ô-( m1 £«m2 )g £½(m1 £«m2 )a (1)
FN2 - m2 g £½m2 a (2)
½âÉÏÊö·½³Ì,µÃ
F£Ô £½(m1 £«m2 )(g £«a) (3)
FN2 £½m2 (g £«a) (4)
(1) µ±Õû¸ö×°ÖÃÒÔ¼ÓËÙ¶Èa £½10 m¡¤£ó ÉÏÉýʱ,ÓÉʽ(3)¿ÉµÃÉþËùÊÜÕÅÁ¦µÄֵΪ
-2
F£Ô £½5.94 ¡Á103 N
ÒÒ¶Ô¼×µÄ×÷ÓÃÁ¦Îª
F¡äN2 £½-FN2 £½-m2 (g £«a) £½-1.98 ¡Á103 N (2) µ±Õû¸ö×°ÖÃÒÔ¼ÓËÙ¶Èa £½1 m¡¤£ó ÉÏÉýʱ,µÃÉþÕÅÁ¦µÄֵΪ
-2
F£Ô £½3.24 ¡Á103 N
´Ëʱ,ÒÒ¶Ô¼×µÄ×÷ÓÃÁ¦ÔòΪ
F¡äN2£½-1.08 ¡Á103 N
ÓÉÉÏÊö¼ÆËã¿É¼û,ÔÚÆðµõÏàͬÖØÁ¿µÄÎïÌåʱ,ÓÉÓÚÆðµõ¼ÓËٶȲ»Í¬,ÉþÖÐËùÊÜÕÅÁ¦Ò²²»Í¬,¼ÓËٶȴó,ÉþÖÐÕÅÁ¦Ò²´ó£®Òò´Ë,ÆðµõÖØÎïʱ±ØÐ뻺Âý¼ÓËÙ,ÒÔÈ·±£Æðµõ¹ý³ÌµÄ°²È«£®
2 -8 Èçͼ(a)Ëùʾ,ÒÑÖªÁ½ÎïÌåA¡¢B µÄÖÊÁ¿¾ùΪm£½3.0kg ÎïÌåA ÒÔ¼ÓËÙ¶Èa £½1.0 m¡¤£ó Ô˶¯,ÇóÎïÌåB Óë×ÀÃæ¼äµÄĦ²ÁÁ¦£®(»¬ÂÖÓëÁ¬½ÓÉþµÄÖÊÁ¿²»¼Æ)
·ÖÎö ¸ÃÌâΪÁ¬½ÓÌåÎÊÌâ,ͬÑù¿ÉÓøôÀëÌå·¨Çó½â£®·ÖÎöʱӦעÒâµ½ÉþÖÐÕÅÁ¦´óС´¦´¦ÏàµÈÊÇÓÐÌõ¼þµÄ,¼´±ØÐëÔÚÉþµÄÖÊÁ¿ºÍÉ쳤¿ÉºöÂÔ¡¢»¬ÂÖÓëÉþÖ®¼äµÄĦ²Á²»¼ÆµÄÇ°ÌáϳÉÁ¢£®Í¬Ê±Ò²Òª×¢Òâµ½ÕÅÁ¦·½ÏòÊDz»Í¬µÄ£®
½â ·Ö±ð¶ÔÎïÌåºÍ»¬ÂÖ×÷ÊÜÁ¦·ÖÎö£Ûͼ(b)£Ý£®ÓÉÅ£¶Ù¶¨ÂÉ·Ö±ð¶ÔÎïÌåA¡¢B ¼°»¬ÂÖÁж¯Á¦Ñ§·½³Ì,ÓÐ
-2
mA g -F£Ô £½mA a (1) F¡ä£Ô1 -F£æ £½mB a¡ä (2) F¡ä£Ô -2F£Ô1 £½0 (3)
¿¼Âǵ½mA £½mB £½m, F£Ô £½F¡ä£Ô , F£Ô1 £½F¡ä£Ô1 ,a¡ä£½2a,¿ÉÁªÁ¢½âµÃÎïÌåÓë×ÀÃæµÄĦ²ÁÁ¦
Ff?
mg??m?4m?a?7.2N 219
Ìâ 2-8 ͼ
ÌÖÂÛ ¶¯Á¦Ñ§ÎÊÌâµÄÒ»°ã½âÌâ²½Öè¿É·ÖΪ£º(1) ·ÖÎöÌâÒâ,È·¶¨Ñо¿¶ÔÏó,·ÖÎöÊÜÁ¦,Ñ¡¶¨×ø±ê£»(2) ¸ù¾ÝÎïÀíµÄ¶¨ÀíºÍ¶¨ÂÉÁгöÔʼ·½³Ì×飻(3) ½â·½³Ì×é,µÃ³öÎÄ×Ö½á¹û£»(4) ºË¶ÔÁ¿¸Ù,ÔÙ´úÈëÊý¾Ý,¼ÆËã³ö½á¹ûÀ´£® 2 -9 ÖÊÁ¿Îªm¡äµÄ³¤Æ½°åA ÒÔËÙ¶Èv¡äÔڹ⻬ƽÃæÉÏ×÷Ö±ÏßÔ˶¯,ÏÖ½«ÖÊÁ¿Îªm µÄľ¿éB ÇáÇáƽÎȵطÅÔÚ³¤Æ½°åÉÏ,°åÓëľ¿éÖ®¼äµÄ¶¯Ä¦²ÁÒòÊýΪ¦Ì,Çóľ¿éÔÚ³¤Æ½°åÉÏ»¬ÐжàÔ¶²ÅÄÜÓë°åÈ¡µÃ¹²Í¬Ëٶȣ¿ ·ÖÎö µ±Ä¾¿éB ƽÎȵØÇáÇá·ÅÖÁÔ˶¯×ŵÄƽ°åA ÉÏʱ,ľ¿éµÄ³õËٶȿÉÊÓΪÁã,ÓÉÓÚËüÓëƽ°åÖ®¼äËٶȵIJîÒì¶ø´æÔÚ»¬¶¯Ä¦²ÁÁ¦,¸ÃÁ¦½«¸Ä±äËüÃǵÄÔ˶¯×´Ì¬£®¸ù¾ÝÅ£¶Ù¶¨Âɿɵõ½ËüÃǸ÷×ÔÏà¶ÔµØÃæµÄ¼ÓËٶȣ®»»ÒÔƽ°åΪ²Î¿¼ÏµÀ´·ÖÎö,´Ëʱ,ľ¿éÒÔ³õËÙ¶È-v¡ä(Óëƽ°åÔ˶¯ËÙÂÊ´óСÏàµÈ¡¢·½ÏòÏà·´)×÷ÔȼõËÙÔ˶¯,Æä¼ÓËÙ¶ÈΪÏà¶Ô¼ÓËÙ¶È,°´Ô˶¯Ñ§¹«Ê½¼´¿É½âµÃ£®
¸ÃÌâÒ²¿ÉÓ¦ÓõÚÈýÕÂËù½²ÊöµÄϵͳµÄ¶¯Äܶ¨ÀíÀ´½â£®½«Æ½°åÓëľ¿é×÷Ϊϵͳ,¸ÃϵͳµÄ¶¯ÄÜÓÉƽ°åÔÓеĶ¯ÄܱäΪľ¿éºÍƽ°åÒ»ÆðÔ˶¯µÄ¶¯ÄÜ,¶øËüÃǵĹ²Í¬Ëٶȿɸù¾Ý¶¯Á¿¶¨ÀíÇóµÃ£®ÓÖÒòΪϵͳÄÚÖ»ÓÐĦ²ÁÁ¦×÷¹¦,¸ù¾ÝϵͳµÄ¶¯Äܶ¨Àí,Ħ²ÁÁ¦µÄ¹¦Ó¦µÈÓÚϵͳ¶¯ÄܵÄÔöÁ¿£®Ä¾¿éÏà¶Ôƽ°åÒƶ¯µÄ¾àÀë¼´¿ÉÇó³ö£®
½â1 ÒÔµØÃæΪ²Î¿¼Ïµ,ÔÚĦ²ÁÁ¦Ff£½¦Ìmg µÄ×÷ÓÃÏÂ,¸ù¾ÝÅ£¶Ù¶¨ÂÉ·Ö±ð¶Ôľ¿é¡¢Æ½°åÁгö¶¯Á¦Ñ§·½³Ì
Ff£½¦Ìmg £½ma
1
Ff£½-Ff£½m¡äa
2
a1 ºÍa2 ·Ö±ðÊÇľ¿éºÍľ°åÏà¶ÔµØÃæ²Î¿¼ÏµµÄ¼ÓËٶȣ®ÈôÒÔľ°åΪ²Î¿¼Ïµ,ľ¿éÏà¶Ôƽ°åµÄ¼ÓËÙ¶Èa £½a1 £«a2 ,ľ¿éÏà¶Ôƽ°åÒÔ³õËÙ¶È- v¡ä×÷ÔȼõËÙÔ˶¯Ö±ÖÁ×îÖÕÍ£Ö¹£®ÓÉÔ˶¯Ñ§¹æÂÉÓÐ
- v¡ä £½2as
ÓÉÉÏÊö¸÷ʽ¿ÉµÃľ¿éÏà¶ÔÓÚƽ°åËùÒƶ¯µÄ¾àÀëΪ
½â2 ÒÔľ¿éºÍƽ°åΪϵͳ,ËüÃÇÖ®¼äÒ»¶ÔĦ²ÁÁ¦×÷µÄ×ܹ¦Îª
2
W?F£¨?Ffl??mgs fs?l£©Ê½ÖÐl Ϊƽ°åÏà¶ÔµØÃæÒƶ¯µÄ¾àÀ룮
ÓÉÓÚϵͳÔÚˮƽ·½ÏòÉϲ»ÊÜÍâÁ¦,µ±Ä¾¿é·ÅÖÁƽ°åÉÏʱ,¸ù¾Ý¶¯Á¿Êغ㶨ÂÉ,ÓÐ
m¡äv¡ä£½(m¡ä£«m) v¡å
ÓÉϵͳµÄ¶¯Äܶ¨Àí,ÓÐ
20
11¦Ìmgs?m?v?2??m??m?v??2
22ÓÉÉÏÊö¸÷ʽ¿ÉµÃ
m?v?2 s??2¦Ìg?m?m?2 -10 Èçͼ(a)Ëùʾ,ÔÚÒ»Ö»°ë¾¶ÎªR µÄ°ëÇòÐÎÍëÄÚ,ÓÐÒ»Á£ÖÊÁ¿Îªm µÄС¸ÖÇò,µ±Ð¡ÇòÒÔ½ÇËٶȦØÔÚˮƽÃæÄÚÑØÍëÄÚ±Ú×÷ÔÈËÙÔ²ÖÜÔ˶¯Ê±,Ëü¾àÍëµ×Óжà¸ß£¿
Ìâ 2-10 ͼ
·ÖÎö ά³Ö¸ÖÇòÔÚˮƽÃæÄÚ×÷ÔȽÇËÙ¶Èת¶¯Ê±,±ØÐëʹ¸ÖÇòÊܵ½Ò»ÓëÏòÐļÓËÙ¶ÈÏà¶ÔÓ¦µÄÁ¦(ÏòÐÄÁ¦),¶ø¸ÃÁ¦ÊÇÓÉÍëÄÚ±Ú¶ÔÇòµÄÖ§³ÖÁ¦FN µÄ·ÖÁ¦À´ÌṩµÄ,ÓÉÓÚÖ§³ÖÁ¦FN ʼÖÕ´¹Ö±ÓÚÍëÄÚ±Ú,ËùÒÔÖ§³ÖÁ¦µÄ´óСºÍ·½ÏòÊÇËæ¦Ø¶ø±äµÄ£®È¡Í¼Ê¾Oxy ×ø±ê,Áгö¶¯Á¦Ñ§·½³Ì,¼´¿ÉÇó½â¸ÖÇò¾àÍëµ×µÄ¸ß¶È£® ½â È¡¸ÖÇòΪ¸ôÀëÌå,ÆäÊÜÁ¦·ÖÎöÈçͼ(b)Ëùʾ£®ÔÚͼʾ×ø±êÖÐÁж¯Á¦Ñ§·½³Ì
FNsin¦È?man?mR¦Ø2sin¦È (1)
FNcos¦È?mg (2)
?R?h? (3)
ÇÒÓÐ cos¦È?RÓÉÉÏÊö¸÷ʽ¿É½âµÃ¸ÖÇò¾àÍëµ×µÄ¸ß¶ÈΪ
h?R?¿É¼û,h Ëæ¦ØµÄ±ä»¯¶ø±ä»¯£®
g¦Ø2
2 -11 ÔÚÈçͼ£¨a£©ËùʾµÄÇỬÂÖÉÏ¿çÓÐÒ»ÇáÉþ£¬ÉþµÄÁ½¶ËÁ¬½Ó×ÅÖÊÁ¿·Ö±ðΪ1 kgºÍ2 kgµÄÎïÌåAºÍB£¬ÏÖÒÔ50 NµÄºãÁ¦FÏòÉÏÌỬÂÖµÄÖᣬ²»¼Æ»¬ÂÖÖÊÁ¿¼°»¬ÂÖÓëÉþ¼äĦ²Á£¬ÇóAºÍBµÄ¼ÓËٶȸ÷Ϊ¶àÉÙ£¿
Ìâ 2-11 ͼ
·ÖÎö ÔÚÉÏÌáÎïÌå¹ý³ÌÖУ¬ÓÉÓÚ»¬ÂÖ¿ÉÒÔת¶¯£¬ËùÒÔA¡¢BÁ½ÎïÌå¶ÔµØ¼ÓËٶȲ¢²»Ïàͬ£¬¹ÊÓ¦½«A¡¢BºÍ»¬ÂÖ·Ö±ð¸ôÀëºó£¬ÔËÓÃÅ£¶Ù¶¨ÂÉÇó½â£¬±¾ÌâÖÐÒò»¬ÂÖÖÊÁ¿¿ÉÒÔ²»¼Æ£¬¹ÊÁ½±ßÉþ×ÓÕÅÁ¦ÏàµÈ£¬ÇÒÓÐF?2FT.
21
½â ¸ôÀëºó£¬¸÷ÎïÌåÊÜÁ¦Èçͼ£¨b£©Ëùʾ£¬ÓÐ »¬ÂÖ FA FTB FTÁªÁ¢Èýʽ£¬µÃ aA ÌÖÂÛ ÈçÓÉʽF?2FT?0
?mAg?mAaA ?mBg?mBaB
?15.2m?s?2£¬aB?2.7m?s?2
?(mA?mB)g?(mA?mB)aÇó½â£¬ËùµÃaÊÇA¡¢BÁ½ÎïÌå¹¹³ÉµÄÖʵãϵµÄÖÊÐÄ
¼ÓËٶȣ¬²¢²»ÊÇA¡¢BÁ½ÎïÌåµÄ¼ÓËÙ¶È.ÉÏʽ½ÐÖÊÐÄÔ˶¯¶¨Àí.
2 -12 Ò»ÖÊÁ¿Îª50 gµÄÎïÌå¹ÒÔÚÒ»µ¯»ÉÄ©¶ËºóÉ쳤һ¶Î¾àÀëºó¾²Ö¹£¬¾ÈŶ¯ºóÎïÌå×÷ÉÏÏÂÕñ¶¯£¬ÈôÒÔÎïÌ徲ƽºâλÖÃΪԵ㣬ÏòÏÂΪyÖáÕýÏò.²âµÃÆäÔ˶¯¹æÂÉ°´ÓàÏÒÐÎʽ¼´
y?0.20cos(5t??/2)£¬Ê½ÖÐtÒÔs
¼Æ£¬yÒÔm¼Æ£¬ÊÔÇ󣺣¨1£©×÷ÓÃÓÚ¸ÃÎïÌåÉϵĺÏÍâÁ¦µÄ´óС£»£¨2£©Ö¤Ã÷×÷ÓÃÔÚÎïÌåÉϵĺÏÍâÁ¦´óСÓëÎïÌåÀ뿪ƽºâλÖõÄy¾àÀë³ÉÕý±È. ·ÖÎö ±¾Ìâ¿ÉÖ±½ÓÓÃF?ma?md2y/dt2Çó½â£¬yΪÎïÌåµÄÔ˶¯·½³Ì£¬F¼´Îª×÷ÓÃÓÚÎïÌåÉϵĺÏÍâÁ¦£¨Êµ
ΪÖØÁ¦Ó뵯»ÉÁ¦Ö®ºÍ£©µÄ±í´ïʽ£¬±¾ÌâÏÔʾÁËÎïÌå×÷¼òгÔ˶¯Ê±µÄ¶¯Á¦Ñ§ÌØÕ÷. ½â £¨1£©ÓÉ·ÖÎöÖª
F?ma?d2y/dt2??0.25cos£¨5t??/2£© £¨N£©
¸Ãʽ±íʾ×÷ÓÃÓÚÎïÌåÉϵĺÏÍâÁ¦Ëæʱ¼ät°´ÓàÏÒ×÷ÓÃÖÜÆÚÐԱ仯£¬F>0±íʾºÏÁ¦ÍâÁ¦ÏòÏ£¬F<0±íʾºÏÍâÁ¦ÏòÉÏ. £¨2£© F??0.25cos(5t??/2)??1.25[0.20cos(5t??/2)]??1.25y.
ÓÉÉÏʽ֪£¬ºÏÍâÁ¦FµÄ´óСÓëÎïÌåÀ뿪ƽºâλÖþàÀëyµÄ´óС³ÉÕý±È.¡°-¡±ºÅ±íʾÓëλÒƵķ½ÏòÏà·´. 2 -13 Ò»ÖÊÁ¿Îª10 kg µÄÖʵãÔÚÁ¦F µÄ×÷ÓÃÏÂÑØx Öá×÷Ö±ÏßÔ˶¯,ÒÑÖªF £½120t £«40,ʽÖÐF µÄµ¥Î»ÎªN,
tµÄµ¥Î»µÄ£ó£®ÔÚt£½0ʱ,ÖʵãλÓÚx £½5.0 m´¦,ÆäËÙ¶Èv0£½6.0 m¡¤s
?1
£®ÇóÖʵãÔÚÈÎÒâʱ¿ÌµÄËٶȺÍλÖã®
·ÖÎö ÕâÊÇÔÚ±äÁ¦×÷ÓÃϵĶ¯Á¦Ñ§ÎÊÌ⣮ÓÉÓÚÁ¦ÊÇʱ¼äµÄº¯Êý,¶ø¼ÓËÙ¶Èa£½dv/dt,Õâʱ,¶¯Á¦Ñ§·½³Ì¾Í³ÉΪËٶȶÔʱ¼äµÄÒ»½×΢·Ö·½³Ì,½â´Ë΢·Ö·½³Ì¿ÉµÃÖʵãµÄËÙ¶Èv (t)£»ÓÉËٶȵĶ¨Òåv£½dx /dt,Óûý·ÖµÄ·½·¨¿ÉÇó³öÖʵãµÄλÖã®
½â Òò¼ÓËÙ¶Èa£½dv/dt,ÔÚÖ±ÏßÔ˶¯ÖÐ,¸ù¾ÝÅ£¶ÙÔ˶¯¶¨ÂÉÓÐ
120t?40?m-1
dv dtÒÀ¾ÝÖʵãÔ˶¯µÄ³õʼÌõ¼þ,¼´t0 £½0 ʱv0 £½6.0 m¡¤£ó ,ÔËÓ÷ÖÀë±äÁ¿·¨¶ÔÉÏʽ»ý·Ö,µÃ
?dv???12.0t?4.0?dt
v00vtv£½6.0+4.0t+6.0t2
ÓÖÒòv£½dx /dt,²¢ÓÉÖʵãÔ˶¯µÄ³õʼÌõ¼þ£ºt0 £½0 ʱ x0 £½5.0 m,¶ÔÉÏʽ·ÖÀë±äÁ¿ºó»ý·Ö,ÓÐ
?xx0dx???6.0?4.0t?6.0t2?dt
t0x £½5.0+6.0t+2.0t2 +2.0t3
2 -14 ÇáÐÍ·É»úÁ¬Í¬¼ÝʻԱ×ÜÖÊÁ¿Îª1.0 ¡Á10 kg£®·É»úÒÔ55.0 m¡¤£ó µÄËÙÂÊÔÚˮƽÅܵÀÉÏ׎ºó,¼Ý
22
3
-1
ʻԱ¿ªÊ¼Öƶ¯,Èô×èÁ¦Óëʱ¼ä³ÉÕý±È,±ÈÀýϵÊý¦Á£½5.0 ¡Á10 N¡¤£ó ,¿ÕÆø¶Ô·É»úÉýÁ¦²»¼Æ,Çó£º(1) 10£óºó·É»úµÄËÙÂÊ£»(2) ·É»ú׎ºó10£óÄÚ»¬ÐеľàÀ룮
·ÖÎö ·É»úÁ¬Í¬¼ÝʻԱÔÚˮƽÅܵÀÉÏÔ˶¯¿ÉÊÓΪÖʵã×÷Ö±ÏßÔ˶¯£®Æäˮƽ·½ÏòËùÊÜÖƶ¯Á¦F Ϊ±äÁ¦,ÇÒÊÇʱ¼äµÄº¯Êý£®ÔÚÇóËÙÂʺ;àÀëʱ,¿É¸ù¾Ý¶¯Á¦Ñ§·½³ÌºÍÔ˶¯Ñ§¹æÂÉ,²ÉÓ÷ÖÀë±äÁ¿·¨Çó½â£®
½â ÒÔµØÃæ·É»ú»¬Ðз½ÏòΪ×ø±êÕý·½Ïò,ÓÉÅ£¶ÙÔ˶¯¶¨Âɼ°³õʼÌõ¼þ,ÓÐF2-1
?ma?mdv??¦Át dt?µÃ vvv0dv???0t¦Átdt m?v0?¦Á2t 2mv £½30 m¡¤£ó-1
Òò´Ë,·É»ú׎10£óºóµÄËÙÂÊΪ
ÓÖ
t?¦Á2?dx?v?dt ??x0?0?02mt??x¹Ê·É»ú׎ºó10£óÄÚËù»¬ÐеľàÀë
s?x?x0?v0t?¦Á3t?467m 6m2
2 -15 ÖÊÁ¿Îªm µÄÌøË®Ô˶¯Ô±,´Ó10.0 m ¸ß̨ÉÏÓɾ²Ö¹ÌøÏÂÂäÈëË®ÖУ®¸ß̨¾àË®Ãæ¾àÀëΪh£®°ÑÌøË®Ô˶¯Ô±ÊÓΪÖʵã,²¢ÂÔÈ¥¿ÕÆø×èÁ¦£®Ô˶¯Ô±ÈëË®ºó´¹Ö±Ï³Á,Ë®¶ÔÆä×èÁ¦Îªbv ,ÆäÖÐb Ϊһ³£Á¿£®ÈôÒÔË®ÃæÉÏÒ»µãΪ×ø±êÔµãO,ÊúÖ±ÏòÏÂΪOy Öá,Çó£º(1) Ô˶¯Ô±ÔÚË®ÖеÄËÙÂÊvÓëy µÄº¯Êý¹Øϵ£»(2) Èçb /m £½0.40m ,ÌøË®Ô˶¯Ô±ÔÚË®ÖÐϳÁ¶àÉÙ¾àÀë²ÅÄÜʹÆäËÙÂÊv¼õÉÙµ½ÂäË®ËÙÂÊv0 µÄ1/10£¿ (¼Ù¶¨ÌøË®Ô˶¯Ô±ÔÚË®Öеĸ¡Á¦ÓëËùÊܵÄÖØÁ¦´óСǡºÃÏàµÈ)
-1
Ìâ 2-15 ͼ
·ÖÎö ¸ÃÌâ¿ÉÒÔ·ÖΪÁ½¸ö¹ý³Ì,ÈëˮǰÊÇ×ÔÓÉÂäÌåÔ˶¯,ÈëË®ºó,ÎïÌåÊÜÖØÁ¦P¡¢¸¡Á¦F ºÍË®µÄ×èÁ¦FfµÄ×÷ÓÃ,ÆäºÏÁ¦ÊÇÒ»±äÁ¦,Òò´Ë,ÎïÌå×÷±ä¼ÓËÙÔ˶¯£®ËäÈ»ÎïÌåµÄÊÜÁ¦·ÖÎö±È½Ï¼òµ¥,µ«ÊÇ,ÓÉÓÚ±äÁ¦ÊÇËٶȵĺ¯Êý(ÔÚÓÐЩÎÊÌâÖбäÁ¦ÊÇʱ¼ä¡¢Î»Öõĺ¯Êý),¶ÔÕâÀàÎÊÌâÁгö¶¯Á¦Ñ§·½³Ì²¢²»¸´ÔÓ,µ«Òª´ÓËü¼ÆËã³öÎïÌåÔ˶¯µÄλÖúÍËٶȾͱȽÏÀ§ÄÑÁË£®Í¨³£ÐèÒª²ÉÓûý·ÖµÄ·½·¨È¥½âËùÁгöµÄ΢·Ö·½³Ì£®ÕâÒ²³ÉÁ˽âÌâ¹ý³ÌÖеÄÄѵ㣮Ôڽⷽ³ÌµÄ¹ý³ÌÖÐ,ÌرðÐèҪעÒâµ½»ý·Ö±äÁ¿µÄͳһºÍ³õʼÌõ¼þµÄÈ·¶¨£® ½â (1) Ô˶¯Ô±Èëˮǰ¿ÉÊÓΪ×ÔÓÉÂäÌåÔ˶¯,¹ÊÈëˮʱµÄËÙ¶ÈΪ
v0?2gh Ô˶¯Ô±ÈëË®ºó,ÓÉÅ£¶Ù¶¨ÂɵÃ
P -Ff -F £½ma
23
ÓÉÌâÒâP £½F¡¢Ff£½bv ,¶øa £½dv /dt £½v (d v /dy),´ú
2
ÈëÉÏʽºóµÃ
-bv2£½ mv (d v /dy)
¿¼Âǵ½³õʼÌõ¼þy0 £½0 ʱ, v0?2gh,¶ÔÉÏʽ»ý·Ö,ÓÐ
vdv?m??dy? ??0??v0v?b?tv?v0e?by/m?2ghe?by/m
(2) ½«ÒÑÖªÌõ¼þb/m £½0.4 m ,v £½0.1v0 ´úÈëÉÏʽ,ÔòµÃ
-1
y??mvln?5.76m bv02 -16 Ò»ÖÊÁ¿Îªm µÄСÇò×î³õλÓÚÈçͼ(a)ËùʾµÄA µã,È»ºóÑذ뾶ΪrµÄ¹â»¬Ô²¹ìµÀADCBÏ»¬£®ÊÔÇóСÇòµ½´ïµãCʱµÄ½ÇËٶȺͶÔÔ²¹ìµÀµÄ×÷ÓÃÁ¦£®
Ìâ 2-16 ͼ
·ÖÎö ¸ÃÌâ¿ÉÓÉÅ£¶ÙµÚ¶þ¶¨ÂÉÇó½â£®ÔÚÈ¡×ÔÈ»×ø±êµÄÇé¿öÏÂ,ÑØÔ²»¡·½ÏòµÄ¼ÓËٶȾÍÊÇÇÐÏò¼ÓËÙ¶Èa£ô,ÓëÆäÏà¶ÔÓ¦µÄÍâÁ¦F£ôÊÇÖØÁ¦µÄÇÐÏò·ÖÁ¿mgsin¦Á,¶øÓë·¨Ïò¼ÓËÙ¶ÈanÏà¶ÔÓ¦µÄÍâÁ¦ÊÇÖ§³ÖÁ¦FN ºÍÖØÁ¦µÄ·¨Ïò·ÖÁ¿
mgcos¦Á£®ÓÉ´Ë,¿É·Ö±ðÁгöÇÐÏòºÍ·¨ÏòµÄ¶¯Á¦Ñ§·½³ÌF£ô£½mdv/dtºÍFn£½man £®ÓÉÓÚСÇòÔÚ»¬¶¯¹ý³ÌÖмÓËÙ¶È
²»ÊǺ㶨µÄ,Òò´Ë,ÐèÓ¦Óûý·ÖÇó½â,ΪʹÔËËã¼ò±ã,¿Éת»»»ý·Ö±äÁ¿£®
¸ÃÌâÒ²ÄÜÓ¦ÓÃÒÔСÇò¡¢Ô²»¡ÓëµØÇòΪϵͳµÄ»úеÄÜÊغ㶨ÂÉÇó½âСÇòµÄËٶȺͽÇËÙ¶È,·½·¨±È½Ï¼ò±ã£®µ«Ëü²»ÄÜÖ±½Ó¸ø³öСÇòÓëÔ²»¡±íÃæÖ®¼äµÄ×÷ÓÃÁ¦£®
½â СÇòÔÚÔ˶¯¹ý³ÌÖÐÊܵ½ÖØÁ¦P ºÍÔ²¹ìµÀ¶ÔËüµÄÖ§³ÖÁ¦FN £®È¡Í¼(b)ËùʾµÄ×ÔÈ»×ø±êϵ,ÓÉÅ£¶Ù¶¨ÂɵÃ
Ft??mgsin¦Á?mdv (1) dt (2)
mv2Fn?FN?mgcos¦Á?mRÓÉv?dsrd¦Árd¦Á,µÃdt?,´úÈëʽ(1),²¢¸ù¾ÝСÇò´ÓµãA Ô˶¯µ½µãC µÄʼĩÌõ¼þ,½øÐлý·Ö,ÓÐ ?vdtdt?µÃ vÔòСÇòÔÚµãC µÄ½ÇËÙ¶ÈΪ
vv0vdv????/2??rgsin??d?
?2rgcos¦Á
24
¦Ø?ÓÉʽ(2)µÃ
v?2gcos¦Á/rr
mv2?mgcos¦Á?3mgcos¦Á FN?mr???FN??3mgcos¦Á FNÓɴ˿ɵÃСÇò¶ÔÔ²¹ìµÀµÄ×÷ÓÃÁ¦Îª
¸ººÅ±íʾF¡äN Óëen ·´Ïò£®
2 -17 ¹â»¬µÄˮƽ×ÀÃæÉÏ·ÅÖÃÒ»°ë¾¶ÎªR µÄ¹Ì¶¨Ô²»·,ÎïÌå½ôÌù»·µÄÄÚ²à×÷Ô²ÖÜÔ˶¯,ÆäĦ²ÁÒòÊýΪ¦Ì,¿ªÊ¼Ê±ÎïÌåµÄËÙÂÊΪv0 ,Çó£º(1) t ʱ¿ÌÎïÌåµÄËÙÂÊ£»(2) µ±ÎïÌåËÙÂÊ´Óv0¼õÉÙv0¼°¾¹ýµÄ·³Ì£®
/2ʱ,ÎïÌåËù¾ÀúµÄʱ¼ä
Ìâ 2-17 ͼ
·ÖÎö Ô˶¯Ñ§Ó붯Á¦Ñ§Ö®¼äµÄÁªÏµÊÇÒÔ¼ÓËÙ¶ÈΪÇÅÁºµÄ,Òò¶ø,¿ÉÏÈ·ÖÎö¶¯Á¦Ñ§ÎÊÌ⣮ÎïÌåÔÚ×÷Ô²ÖÜÔ˶¯µÄ¹ý³ÌÖÐ,´ÙʹÆäÔ˶¯×´Ì¬·¢Éú±ä»¯µÄÊÇÔ²»·ÄÚ²à¶ÔÎïÌåµÄÖ§³ÖÁ¦FN ºÍ»·ÓëÎïÌåÖ®¼äµÄĦ²ÁÁ¦F£æ ,¶øĦ²ÁÁ¦´óСÓëÕýѹÁ¦FN¡ä³ÉÕý±È,ÇÒFNÓëFN¡äÓÖÊÇ×÷ÓÃÁ¦Óë·´×÷ÓÃÁ¦,ÕâÑù,¾Í¿Éͨ¹ýËüÃÇ°ÑÇÐÏòºÍ·¨ÏòÁ½¸ö¼ÓËÙ¶ÈÁªÏµÆðÀ´ÁË,´Ó¶ø¿ÉÓÃÔ˶¯Ñ§µÄ»ý·Ö¹ØϵʽÇó½âËÙÂʺÍ·³Ì£® ½â (1) ÉèÎïÌåÖÊÁ¿Îªm,ȡͼÖÐËùʾµÄ×ÔÈ»×ø±ê,°´Å£¶Ù¶¨ÂÉ,ÓÐ
mv2FN?man?
RFf??mat??ÓÉ·ÖÎöÖпÉÖª,Ħ²ÁÁ¦µÄ´óСF£æ£½¦ÌFN ,ÓÉÉÏÊö¸÷ʽ¿ÉµÃ
dv dtv2dv¦Ì?? RdtÈ¡³õʼÌõ¼þt £½0 ʱv £½v 0 ,²¢¶ÔÉÏʽ½øÐлý·Ö,ÓÐ
?t0dt??Rvdv
¦Ì?v0v2
v?Rv0R?v0¦Ìt 25
(2) µ±ÎïÌåµÄËÙÂÊ´Óv 0 ¼õÉÙµ½v0/2ʱ,ÓÉÉÏʽ¿ÉµÃËùÐèµÄʱ¼äΪ
t??ÎïÌåÔÚÕâ¶Îʱ¼äÄÚËù¾¹ýµÄ·³Ì
R¦Ìv0
s??vdt??0t?t?0Rv0dt
R?v0¦Ìts?Rln2 ¦Ì2
2 -18 Ò»ÎïÌå×ÔµØÇò±íÃæÒÔËÙÂÊv0 ÊúÖ±ÉÏÅ×£®¼Ù¶¨¿ÕÆø¶ÔÎïÌå×èÁ¦µÄֵΪFr £½kmv ,ÆäÖÐm ΪÎïÌåµÄÖÊÁ¿,k Ϊ³£Á¿£®ÊÔÇó£º(1) ¸ÃÎïÌåÄÜÉÏÉýµÄ¸ß¶È£»(2)ÎïÌå·µ»ØµØÃæʱËٶȵÄÖµ£®(ÉèÖØÁ¦¼ÓËÙ¶ÈΪ³£Á¿£®)
Ìâ 2-18 ͼ
·ÖÎö ÓÉÓÚ¿ÕÆø¶ÔÎïÌåµÄ×èÁ¦Ê¼ÖÕÓëÎïÌåÔ˶¯µÄ·½ÏòÏà·´,Òò´Ë,ÎïÌåÔÚÉÏÅ×¹ý³ÌÖÐËùÊÜÖØÁ¦PºÍ×èÁ¦Fr µÄ·½ÏòÏàͬ£»¶øÏÂÂä¹ý³ÌÖÐ,ËùÊÜÖØÁ¦P ºÍ×èÁ¦Fr µÄ·½ÏòÔòÏà·´£®ÓÖÒò×èÁ¦ÊDZäÁ¦,Ôڽ⶯Á¦Ñ§·½³Ìʱ,ÐèÓûý·ÖµÄ·½·¨£®
½â ·Ö±ð¶ÔÎïÌåÉÏÅס¢ÏÂÂäʱ×÷ÊÜÁ¦·ÖÎö,ÒÔµØÃæΪԵã,ÊúÖ±ÏòÉÏΪy Öá(ÈçͼËùʾ)£®(1) ÎïÌåÔÚÉÏÅ×¹ý³ÌÖÐ,¸ù¾ÝÅ£¶Ù¶¨ÂÉÓÐ
?mg?kmv2?mÒÀ¾Ý³õʼÌõ¼þ¶ÔÉÏʽ»ý·Ö,ÓÐ
dvvdv?m dtdy?y0dy???0v0vdvg?kv2
1?g?kv2?? y??ln?2??2k?g?kv0?ÎïÌåµ½´ï×î¸ß´¦Ê±, v £½0,¹ÊÓÐ
2?1?g?kv0? h?ymax?ln???2k?g? 26
(2) ÎïÌåÏÂÂä¹ý³ÌÖÐ,ÓÐ
?mg?kmv2?m¶ÔÉÏʽ»ý·Ö,ÓÐ
vdv dy?y0dy????1/20v0vdvg?kv2
?kv2?Ôò v?v0??1?g????
2 -19 ÖÊÁ¿Îªm µÄĦÍгµ,Ôں㶨µÄÇ£ÒýÁ¦FµÄ×÷ÓÃϹ¤×÷,ËüËùÊܵÄ×èÁ¦ÓëÆäËÙÂʵÄƽ·½³ÉÕý±È,ËüÄÜ´ïµ½µÄ×î´óËÙÂÊÊÇvm £®ÊÔ¼ÆËã´Ó¾²Ö¹¼ÓËÙµ½vm/2ËùÐèµÄʱ¼äÒÔ¼°Ëù×ß¹ýµÄ·³Ì£®
·ÖÎö ¸ÃÌâÒÀÈ»ÊÇÔËÓö¯Á¦Ñ§·½³ÌÇó½â±äÁ¦×÷ÓÃϵÄËٶȺÍλÖõÄÎÊÌâ,Çó½â·½·¨ÓëÇ°Á½ÌâÏàËÆ,Ö»ÊÇÔÚ½âÌâ¹ý³ÌÖбØÐëÉè·¨Çó³ö×èÁ¦ÏµÊýk£®ÓÉÓÚ×èÁ¦Fr£½kv ,ÇÒFrÓÖÓëºãÁ¦F µÄ·½ÏòÏà·´£»¹Êµ±×èÁ¦ËæËÙ¶ÈÔö¼ÓÖÁÓëºãÁ¦´óСÏàµÈʱ,¼ÓËÙ¶ÈΪÁã,´ËʱËٶȴﵽ×î´ó£®Òò´Ë,¸ù¾ÝËÙ¶È×î´óÖµ¿ÉÇó³ö×èÁ¦ÏµÊýÀ´£®µ«ÔÚÇóĦÍгµËù×ß·³Ìʱ,Ðè¶Ô±äÁ¿×÷±ä»»£®
½â ÉèĦÍгµÑØx ÖáÕý·½ÏòÔ˶¯,ÔÚÇ£ÒýÁ¦FºÍ×èÁ¦Fr ͬʱ×÷ÓÃÏÂ,ÓÉÅ£¶Ù¶¨ÂÉÓÐ
2
F?kv2?mµ±¼ÓËÙ¶Èa £½dv/dt £½0 ʱ,ĦÍгµµÄËÙÂÊ×î´ó,Òò´Ë¿ÉµÃ
dv (1) dtk£½F/vm2 (2)
ÓÉʽ(1)ºÍʽ(2)¿ÉµÃ
?v2?dv? (3) F?1??m?v2?dtm??¸ù¾ÝʼĩÌõ¼þ¶Ôʽ(3)»ý·Ö,ÓÐ
?1?Ôò tÓÖÒòʽ(3)ÖÐmt0dt?mF?1vm20?v2???1?v2??dv
m???mvmln3 2Fdvmvdv,ÔÙÀûÓÃʼĩÌõ¼þ¶Ôʽ(3)»ý·Ö,ÓÐ ?dtdx?x0dx?mF?1vm20?v2???1?v2??dv
m??
?122mvm4mvmln?0.144Ôò x?2F3F
*2 -20 ÔÚ¿¨³µ³µÏáµ×°åÉÏ·ÅһľÏä,¸ÃľÏä¾à³µÏäÇ°Ñص²°åµÄ¾àÀëL £½2.0 m,ÒÑÖªÖƶ¯Ê±¿¨³µµÄ¼ÓËÙ¶Èa
27
£½7.0 m¡¤£ó ,ÉèÖƶ¯Ò»¿ªÊ¼Ä¾Ïä¾Í¿ªÊ¼»¬¶¯£®Çó¸ÃľÏäײÉϵ²°åʱÏà¶Ô¿¨³µµÄËÙÂÊΪ¶à´ó£¿ÉèľÏäÓëµ×°å¼ä»¬¶¯Ä¦²ÁÒòÊý¦Ì£½0.50£®
·ÖÎö ÈçͬϰÌâ2 -5 ·ÖÎöÖÐÖ¸³öµÄÄÇÑù,¿É¶ÔľÏä¼ÓÉϹßÐÔÁ¦F0 ºó,ÒÔ³µÏáΪ²Î¿¼Ïµ½øÐÐÇó½â,ÈçͼËùʾ,´ËʱľÏäÔÚˮƽ·½ÏòÊܵ½¹ßÐÔÁ¦ºÍĦ²ÁÁ¦×÷ÓÃ,ͼÖÐa¡äΪľÏäÏà¶Ô³µÏáµÄ¼ÓËٶȣ® ½â ÓÉÅ£¶ÙµÚ¶þ¶¨ÂɺÍÏà¹ØÔ˶¯Ñ§¹æÂÉÓÐ
-2
F0 -Ff£½ma -¦Ìmg£½ma¡ä (1) v¡ä 2 £½2a¡äL (2)
ÁªÁ¢½â(1)(2)Á½Ê½²¢´úÈëÌâ¸øÊý¾Ý,µÃľÏäײÉϳµÏáµ²°åʱµÄËÙ¶ÈΪ
v??2?a??g?L=?2.9m?s?1
µÚÈýÕ ¶¯Á¿Êغ㶨ÂɺÍÄÜÁ¿Êغ㶨ÂÉ
3 -1 ¶ÔÖʵã×éÓÐÒÔϼ¸ÖÖ˵·¨£º (1) Öʵã×é×ܶ¯Á¿µÄ¸Ä±äÓëÄÚÁ¦Î޹أ» (2) Öʵã×é×ܶ¯ÄܵĸıäÓëÄÚÁ¦Î޹أ» (3) Öʵã×é»úеÄܵĸıäÓë±£ÊØÄÚÁ¦Î޹أ® ÏÂÁжÔÉÏÊö˵·¨ÅжÏÕýÈ·µÄÊÇ( )
(A) Ö»ÓÐ(1)ÊÇÕýÈ·µÄ (B) (1)¡¢(2)ÊÇÕýÈ·µÄ (C) (1)¡¢(3)ÊÇÕýÈ·µÄ (D) (2)¡¢(3)ÊÇÕýÈ·µÄ
·ÖÎöÓë½â ÔÚÖʵã×éÖÐÄÚÁ¦×ÜÊdzɶԳöÏÖµÄ,ËüÃÇÊÇ×÷ÓÃÁ¦Óë·´×÷ÓÃÁ¦£®ÓÉÓÚÒ»¶ÔÄÚÁ¦µÄ³åÁ¿ºãΪÁã,¹ÊÄÚÁ¦²»»á¸Ä±äÖʵã×éµÄ×ܶ¯Á¿£®µ«ÓÉÓÚÏ໥ÓÐ×÷ÓÃÁ¦µÄÁ½¸öÖʵãµÄλÒÆ´óСÒÔ¼°Î»ÒÆÓëÁ¦µÄ¼Ð½ÇÒ»°ã²»Í¬,¹ÊÒ»¶ÔÄÚÁ¦Ëù×÷¹¦Ö®ºÍ²»Ò»¶¨ÎªÁã,Ó¦×÷¾ßÌå·ÖÎö,ÈçÒ»¶Ôµ¯ÐÔÄÚÁ¦µÄ¹¦µÄ´úÊýºÍÒ»°ãΪÁã,Ò»¶ÔĦ²ÁÄÚÁ¦µÄ¹¦´úÊýºÍÒ»°ã²»ÎªÁã,¶ÔÓÚ±£ÊØÄÚÁ¦À´Ëµ,Ëù×÷¹¦ÄÜʹÖʵã×鶯ÄÜÓëÊÆÄÜÏ໥ת»»,Òò´Ë±£ÊØÄÚÁ¦¼´Ê¹ÓпÉÄܸıäÖʵã×éµÄ¶¯ÄÜ,µ«Ò²²»¿ÉÄܸıäÖʵã×éµÄ»úеÄÜ£®×ÛÉÏËùÊö(1)(3)˵·¨ÊÇÕýÈ·µÄ£®¹ÊÑ¡(C)£® 3 -2 ÓÐÁ½¸öÇã½Ç²»Í¬¡¢¸ß¶ÈÏàͬ¡¢ÖÊÁ¿Ò»ÑùµÄбÃæ·ÅÔڹ⻬µÄˮƽÃæÉÏ,бÃæÊǹ⻬µÄ,ÓÐÁ½¸öÒ»ÑùµÄÎï¿é·Ö±ð´ÓÕâÁ½¸öбÃæµÄ¶¥µãÓɾ²Ö¹¿ªÊ¼»¬ÏÂ,Ôò( ) (A) Îï¿éµ½´ïбÃæµ×¶ËʱµÄ¶¯Á¿ÏàµÈ (B) Îï¿éµ½´ïбÃæµ×¶Ëʱ¶¯ÄÜÏàµÈ
(C) Îï¿éºÍбÃæ(ÒÔ¼°µØÇò)×é³ÉµÄϵͳ,»úеÄܲ»Êغã (D) Îï¿éºÍбÃæ×é³ÉµÄϵͳˮƽ·½ÏòÉ϶¯Á¿Êغã
·ÖÎöÓë½â ¶ÔÌâÊöϵͳÀ´Ëµ,ÓÉÌâÒâÖª²¢ÎÞÍâÁ¦ºÍ·Ç±£ÊØÄÚÁ¦×÷¹¦,¹Êϵͳ»úеÄÜÊغ㣮ÎïÌåÔÚÏ»¬¹ý³ÌÖÐ,Ò»·½Ãæͨ¹ýÖØÁ¦×÷¹¦½«ÊÆÄÜת»¯Îª¶¯ÄÜ,ÁíÒ»·½Ãæͨ¹ýÎïÌåÓëбÃæÖ®¼äµÄµ¯ÐÔÄÚÁ¦×÷¹¦½«Ò»²¿·ÖÄÜÁ¿×ª»¯ÎªÐ±ÃæµÄ¶¯ÄÜ,Æä´óСȡ¾öÆäÖÐÒ»¸öÄÚÁ¦Ëù×÷¹¦£®ÓÉÓÚбÃæÇã½Ç²»Í¬,¹ÊÎïÌåÑز»Í¬Çã½ÇбÃ滬ÖÁµ×¶Ëʱ¶¯ÄÜ´óС²»µÈ£®¶¯Á¿×ÔȻҲ¾Í²»µÈ(¶¯Á¿·½ÏòÒ²²»Í¬)£®¹Ê(A)(B)(C)ÈýÖÖ˵·¨¾ù²»ÕýÈ·£®ÖÁÓÚ˵·¨(D)ÕýÈ·,
28
ÊÇÒòΪ¸Ãϵͳ¶¯Á¿Ëä²»Êغã(Ï»¬Ç°ÏµÍ³¶¯Á¿ÎªÁã,Ï»¬ºóÎïÌåÓëбÃ涯Á¿µÄʸÁ¿ºÍ²»¿ÉÄÜΪÁ㣮ÓÉ´Ë¿ÉÖª,´ËʱÏòÉϵĵØÃæÖ§³ÖÁ¦²¢²»µÈÓÚÎïÌåÓëбÃæÏòϵÄÖØÁ¦),µ«ÔÚˮƽ·½ÏòÉϲ¢ÎÞÍâÁ¦,¹ÊϵͳÔÚˮƽ·½ÏòÉÏ·Ö¶¯Á¿Êغ㣮
3 -3 ¶Ô¹¦µÄ¸ÅÄîÓÐÒÔϼ¸ÖÖ˵·¨£º
(1) ±£ÊØÁ¦×÷Õý¹¦Ê±,ϵͳÄÚÏàÓ¦µÄÊÆÄÜÔö¼Ó£» (2) ÖʵãÔ˶¯¾Ò»±ÕºÏ·¾¶,±£ÊØÁ¦¶ÔÖʵã×÷µÄ¹¦ÎªÁ㣻
(3) ×÷ÓÃÁ¦ºÍ·´×÷ÓÃÁ¦´óСÏàµÈ¡¢·½ÏòÏà·´,ËùÒÔÁ½ÕßËù×÷¹¦µÄ´úÊýºÍ±ØΪÁ㣮 ÏÂÁÐÉÏÊö˵·¨ÖÐÅжÏÕýÈ·µÄÊÇ( )
(A) (1)¡¢(2)ÊÇÕýÈ·µÄ (B) (2)¡¢(3)ÊÇÕýÈ·µÄ (C) Ö»ÓÐ(2)ÊÇÕýÈ·µÄ(D) Ö»ÓÐ(3)ÊÇÕýÈ·µÄ
·ÖÎöÓë½â ±£ÊØÁ¦×÷Õý¹¦Ê±,ϵͳÄÚÏàÓ¦ÊÆÄÜÓ¦¸Ã¼õÉÙ£®ÓÉÓÚ±£ÊØÁ¦×÷¹¦Óë·¾¶ÎÞ¹Ø,¶øÖ»ÓëʼĩλÖÃÓйØ,ÈçÖʵ㻷ÈÆÒ»Öܹý³ÌÖÐ,±£ÊØÁ¦ÔÚÒ»¶Î¹ý³ÌÖÐ×÷Õý¹¦,ÔÚÁíÒ»¶Î¹ý³ÌÖбØÈ»×÷¸º¹¦,Á½ÕßÖ®ºÍ±ØΪÁ㣮ÖÁÓÚÒ»¶Ô×÷ÓÃÁ¦Óë·´×÷ÓÃÁ¦·Ö±ð×÷ÓÃÓÚÁ½¸öÖʵãËù×÷¹¦Ö®ºÍδ±ØΪÁã(Ïê¼ûÏ°Ìâ3 -2 ·ÖÎö),Óɴ˿ɼûÖ»ÓÐ˵·¨(2)ÕýÈ·,¹ÊÑ¡(C)£®
3 -4 ÈçͼËùʾ,ÖÊÁ¿·Ö±ðΪm1 ºÍm2 µÄÎïÌåA ºÍB,ÖÃÓڹ⻬×ÀÃæÉÏ,A ºÍB Ö®¼äÁ¬ÓÐÒ»Çᵯ»É£®ÁíÓÐÖÊÁ¿Îª
m1 ºÍm2 µÄÎïÌåC ºÍD ·Ö±ðÖÃÓÚÎïÌåA ÓëB Ö®ÉÏ,ÇÒÎïÌåAºÍC¡¢B ºÍD Ö®¼äµÄĦ²ÁÒòÊý¾ù²»ÎªÁ㣮Ê×ÏÈÓÃÍâ
Á¦ÑØˮƽ·½ÏòÏàÏòÍÆѹA ºÍB,ʹµ¯»É±»Ñ¹Ëõ,È»ºó³·µôÍâÁ¦,ÔòÔÚAºÍB µ¯¿ªµÄ¹ý³ÌÖÐ,¶ÔA¡¢B¡¢C¡¢D ÒÔ¼°µ¯»É×é³ÉµÄϵͳ,ÓÐ( )
(A) ¶¯Á¿Êغã,»úеÄÜÊغã (B) ¶¯Á¿²»Êغã,»úеÄÜÊغã (C) ¶¯Á¿²»Êغã,»úеÄܲ»Êغã (D) ¶¯Á¿Êغã,»úеÄܲ»Ò»¶¨Êغã
·ÖÎöÓë½â ÓÉÌâÒâÖª,×÷ÓÃÔÚÌâÊöϵͳÉϵĺÏÍâÁ¦ÎªÁã,¹Êϵͳ¶¯Á¿Êغã,µ«»úеÄÜδ±ØÊغã,ÕâÈ¡¾öÓÚÔÚA¡¢B µ¯¿ª¹ý³ÌÖÐC ÓëA »òD ÓëB Ö®¼äÓÐÎÞÏà¶Ô»¬¶¯,ÈçÓÐÔò±ØÈ»»áÒòĦ²ÁÄÚÁ¦×÷¹¦,¶øʹһ²¿·Ö»úеÄÜת»¯ÎªÈÈÄÜ,¹ÊÑ¡(D)£®
3 -5 ÈçͼËùʾ,×Óµ¯ÉäÈë·ÅÔÚˮƽ¹â»¬µØÃæÉϾ²Ö¹µÄľ¿éºó¶ø´©³ö£®ÒÔµØÃæΪ²Î¿¼Ïµ,ÏÂÁÐ˵·¨ÖÐÕýÈ·µÄ˵·¨ÊÇ( )
(A) ×Óµ¯¼õÉٵĶ¯ÄÜת±äΪľ¿éµÄ¶¯ÄÜ (B) ×Óµ¯-ľ¿éϵͳµÄ»úеÄÜÊغã
(C) ×Óµ¯¶¯ÄܵļõÉÙµÈÓÚ×Óµ¯¿Ë·þľ¿é×èÁ¦Ëù×÷µÄ¹¦ (D) ×Óµ¯¿Ë·þľ¿é×èÁ¦Ëù×÷µÄ¹¦µÈÓÚÕâÒ»¹ý³ÌÖвúÉúµÄÈÈ
29
·ÖÎöÓë½â ×Óµ¯-ľ¿éϵͳÔÚ×Óµ¯ÉäÈë¹ý³ÌÖÐ,×÷ÓÃÓÚϵͳµÄºÏÍâÁ¦ÎªÁã,¹Êϵͳ¶¯Á¿Êغã,µ«»úеÄܲ¢²»Êغ㣮ÕâÊÇÒòΪ×Óµ¯Óëľ¿é×÷ÓõÄÒ»¶ÔÄÚÁ¦Ëù×÷¹¦µÄ´úÊýºÍ²»ÎªÁã(ÕâÊÇÒòΪ×Óµ¯¶ÔµØλÒÆ´óÓÚľ¿é¶ÔµØλÒÆËùÖÂ),×Óµ¯¶¯ÄܵļõÉÙµÈÓÚ×Óµ¯¿Ë·þ×èÁ¦Ëù×÷¹¦,×Óµ¯¼õÉٵĶ¯ÄÜÖÐ,Ò»²¿·Öͨ¹ýÆä·´×÷ÓÃÁ¦¶Ôľ¿é×÷Õý¹¦¶øתÒÆΪľ¿éµÄ¶¯ÄÜ,ÁíÒ»²¿·ÖÔòת»¯ÎªÈÈÄÜ(´óС¾ÍµÈÓÚÕâÒ»¶ÔÄÚÁ¦Ëù×÷¹¦µÄ´úÊýºÍ)£®×ÛÉÏËùÊö,Ö»ÓÐ˵·¨(C)µÄ±íÊöÊÇÍêÈ«ÕýÈ·µÄ£®
3 -6 Ò»¼ÜÒÔ3.0 ¡Á10m¡¤£ó µÄËÙÂÊˮƽ·ÉÐеķɻú,ÓëÒ»Ö»Éí³¤Îª0.20 m¡¢ÖÊÁ¿Îª0.50 kg µÄ·ÉÄñÏàÅö£®ÉèÅöײºó·ÉÄñµÄʬÌåÓë·É»ú¾ßÓÐͬÑùµÄËÙ¶È,¶øÔÀ´·ÉÄñ¶ÔÓÚµØÃæµÄËÙÂÊÉõС,¿ÉÒÔºöÂÔ²»¼Æ£®ÊÔ¹À¼Æ·ÉÄñ¶Ô·É»úµÄ³å»÷Á¦(Åöײʱ¼ä¿ÉÓ÷ÉÄñÉí³¤±»·É»úËÙÂÊÏà³ýÀ´¹ÀËã)£®¸ù¾Ý±¾ÌâµÄ¼ÆËã½á¹û,Äã¶ÔÓÚ¸ßËÙÔ˶¯µÄÎïÌå(Èç·É»ú¡¢Æû³µ)Óëͨ³£Çé¿öϲ»×ãÒÔÒýÆðΣº¦µÄÎïÌå(Èç·ÉÄñ¡¢Ð¡Ê¯×Ó)ÏàÅöºó»á²úÉúʲôºó¹ûµÄÎÊÌâÓÐЩʲôÌå»á£¿
·ÖÎö ÓÉÓÚÄñÓë·É»úÖ®¼äµÄ×÷ÓÃÊÇÒ»¶ÌÔÝʱ¼äÄÚ¼±¾ç±ä»¯µÄ±äÁ¦,Ö±½ÓÓ¦ÓÃÅ£¶Ù¶¨Âɽâ¾öÊÜÁ¦ÎÊÌâÊDz»¿ÉÄܵģ®Èç¹û¿¼ÂÇÁ¦µÄʱ¼äÀÛ»ýЧ¹û,ÔËÓö¯Á¿¶¨ÀíÀ´·ÖÎö,¾Í¿É±ÜÃâ×÷Óùý³ÌÖеÄϸ½ÚÇé¿ö£®ÔÚÇóÄñ¶Ô·É»úµÄ³åÁ¦(³£Ö¸ÔÚ¶ÌÔÝʱ¼äÄÚµÄƽ¾ùÁ¦)ʱ,ÓÉÓÚ·É»úµÄ״̬(Ö¸¶¯Á¿)±ä»¯²»ÖªµÀ,ʹ¼ÆËãÒ²ÄÑÒÔ½øÐУ»Õâʱ,¿É½«ÎÊÌâת»¯ÎªÌÖÂÛÄñµÄ״̬±ä»¯À´·ÖÎöÆäÊÜÁ¦Çé¿ö,²¢¸ù¾ÝÄñÓë·É»ú×÷ÓõÄÏ໥ÐÔ(×÷ÓÃÓë·´×÷ÓÃ),ÎÊÌâ¾ÍºÜ¼òµ¥ÁË£®
½â ÒÔ·ÉÄñΪÑо¿¶ÔÏó,È¡·É»úÔ˶¯·½ÏòΪx ÖáÕýÏò£®Óɶ¯Á¿¶¨ÀíµÃ
2-1
F?¦¤t?mv?0 ʽÖÐF?Ϊ·É»ú¶ÔÄñµÄƽ¾ù³åÁ¦,¶øÉí³¤Îª20cm µÄ·ÉÄñÓë·É»úÅöײʱ¼äԼΪ¦¤t £½l /v,ÒÔ´Ë´úÈëÉÏʽ¿ÉµÃ
mv2F???2.55?105N
lÄñ¶Ô·É»úµÄƽ¾ù³åÁ¦Îª
F??F???2.55?105N
ʽÖиººÅ±íʾ·É»úÊܵ½µÄ³åÁ¦ÓëÆä·ÉÐз½ÏòÏà·´£®´Ó¼ÆËã½á¹û¿ÉÖª,2.25 ¡Á10 NµÄ³åÁ¦´óÖÂÏ൱ÓÚÒ»¸ö22 t µÄÎïÌåËùÊܵÄÖØÁ¦,¿É¼û,´Ë³åÁ¦ÊÇÏ൱´óµÄ£®Èô·ÉÄñÓë·¢¶¯»úҶƬÏàÅö,×ãÒÔʹ·¢¶¯»úËð»µ,Ôì³É·ÉÐÐʹʣ®
3 -7 ÈçͼËùʾ£¬ÖÊÁ¿Îªm µÄÎïÌå,ÓÉˮƽÃæÉϵãO ÒÔ³õËÙΪv0 Å׳ö,v0ÓëˮƽÃæ³ÉÑö½Ç¦Á£®Èô²»¼Æ¿ÕÆø×èÁ¦,Çó£º(1) ÎïÌå´Ó·¢ÉäµãO µ½×î¸ßµãµÄ¹ý³ÌÖÐ,ÖØÁ¦µÄ³åÁ¿£»(2) ÎïÌå´Ó·¢Éäµãµ½Âä»ØÖÁͬһˮƽÃæµÄ¹ý³ÌÖÐ,ÖØÁ¦µÄ³åÁ¿£®
5
30
·ÖÎö ÖØÁ¦ÊǺãÁ¦,Òò´Ë,ÇóÆäÔÚÒ»¶Îʱ¼äÄڵijåÁ¿Ê±,Ö»ÐèÇó³öʱ¼ä¼ä¸ô¼´¿É£®ÓÉÅ×ÌåÔ˶¯¹æÂÉ¿ÉÖª,ÎïÌåµ½´ï×î¸ßµãµÄʱ¼ä¦¤t1?v0sin¦Ág,ÎïÌå´Ó³ö·¢µ½Âä»ØÖÁͬһˮƽÃæËùÐèµÄʱ¼äÊǵ½´ï×î¸ßµãʱ¼äµÄÁ½
±¶£®ÕâÑù,°´³åÁ¿µÄ¶¨Òå¼´¿ÉÇóµÃ½á¹û£®
ÁíÒ»ÖÖ½âµÄ·½·¨ÊǸù¾Ý¹ý³ÌµÄʼ¡¢Ä©¶¯Á¿,Óɶ¯Á¿¶¨ÀíÇó³ö£® ½â1 ÎïÌå´Ó³ö·¢µ½´ï×î¸ßµãËùÐèµÄʱ¼äΪ
¦¤t1?ÔòÎïÌåÂä»ØµØÃæµÄʱ¼äΪ
v0sin¦Á g¦¤t2?2¦¤t1?ÓÚÊÇ,ÔÚÏàÓ¦µÄ¹ý³ÌÖÐÖØÁ¦µÄ³åÁ¿·Ö±ðΪ
2v0sin?g
I1??¦¤t1Fdt??mg¦¤t1j??mv0sin?j
I2??¦¤t2Fdt??mg¦¤t2j??2mv0sin?j
½â2 ¸ù¾Ý¶¯Á¿¶¨Àí,ÎïÌåÓÉ·¢ÉäµãO Ô˶¯µ½µãA¡¢B µÄ¹ý³ÌÖÐ,ÖØÁ¦µÄ³åÁ¿·Ö±ðΪ
I1?mvAyj?mv0yj??mv0sin¦Áj I2?mvByj?mv0yj??2mv0sin¦Áj
3 -8 Fx £½30£«4t(ʽÖÐFx µÄµ¥Î»ÎªN,t µÄµ¥Î»Îªs)µÄºÏÍâÁ¦×÷ÓÃÔÚÖÊÁ¿m£½10 kg µÄÎïÌåÉÏ,ÊÔÇó£º(1) ÔÚ¿ªÊ¼2£ó ÄÚ´ËÁ¦µÄ³åÁ¿£»(2) Èô³åÁ¿I £½300 N¡¤s,´ËÁ¦×÷ÓõÄʱ¼ä£»(3) ÈôÎïÌåµÄ³õËÙ¶Èv1 £½10 m¡¤s ,·½ÏòÓëFxÏàͬ,ÔÚt£½6.86 sʱ,´ËÎïÌåµÄËÙ¶Èv2 £® ·ÖÎö ±¾Ìâ¿ÉÓɳåÁ¿µÄ¶¨ÒåʽI½â (1) ÓÉ·ÖÎöÖª
-1
??tFdt,Çó±äÁ¦µÄ³åÁ¿,¼Ì¶ø¸ù¾Ý¶¯Á¿¶¨ÀíÇóÎïÌåµÄËÙ¶Èv£®
2
t21I??0?30?4t?dt?30t?2t2(2) ÓÉI £½300 £½30t £«2t ,½â´Ë·½³Ì¿ÉµÃ
2
220?68N?s
t £½6.86 s(ÁíÒ»½â²»ºÏÌâÒâÒÑÉáÈ¥)
31
(3) Óɶ¯Á¿¶¨Àí,ÓÐ
I £½m v2- m v1
ÓÉ(2)¿ÉÖªt £½6.86 s ʱI £½300 N¡¤s ,½«I¡¢m ¼°v1´úÈë¿ÉµÃ
v2?I?mv1?40m?s?1 m3 -9 ¸ß¿Õ×÷ҵʱϵ°²È«´øÊǷdz£±ØÒªµÄ£®¼ÙÈçÒ»ÖÊÁ¿Îª51.0 kg µÄÈË,ÔÚ²Ù×÷ʱ²»É÷´Ó¸ß¿ÕÊúÖ±µøÂäÏÂÀ´,ÓÉÓÚ°²È«´øµÄ±£»¤,×îÖÕʹËû±»Ðü¹ÒÆðÀ´£®ÒÑÖª´ËʱÈËÀëÔ´¦µÄ¾àÀëΪ2.0 m ,°²È«´øµ¯ÐÔ»º³å×÷ÓÃʱ¼äΪ0.50 s £®Çó°²È«´ø¶ÔÈ˵Äƽ¾ù³åÁ¦£®
·ÖÎö ´ÓÈËÊÜÁ¦µÄÇé¿öÀ´¿´,¿É·ÖÁ½¸ö½×¶Î£ºÔÚ¿ªÊ¼ÏÂÂäµÄ¹ý³ÌÖÐ,Ö»ÊÜÖØÁ¦×÷ÓÃ,ÈËÌå¿É¿´³ÉÊÇ×÷×ÔÓÉÂäÌåÔ˶¯£»ÔÚ°²È«´ø±£»¤µÄ»º³å¹ý³ÌÖÐ,ÔòÈËÌåͬʱÊÜÖØÁ¦ºÍ°²È«´ø³åÁ¦µÄ×÷ÓÃ,ÆäºÏÁ¦ÊÇÒ»±äÁ¦,ÇÒ×÷ÓÃʱ¼äºÜ¶Ì£®ÎªÇó°²È«´øµÄ³åÁ¦,¿ÉÒÔ´Ó»º³åʱ¼äÄÚ,ÈËÌåÔ˶¯×´Ì¬(¶¯Á¿)µÄ¸Ä±äÀ´·ÖÎö,¼´ÔËÓö¯Á¿¶¨ÀíÀ´ÌÖÂÛ£®ÊÂʵÉÏ,¶¯Á¿¶¨ÀíÒ²¿ÉÓ¦ÓÃÓÚÕû¸ö¹ý³Ì£®µ«ÊÇ,Õâʱ±ØÐë·ÖÇåÖØÁ¦ºÍ°²È«´ø³åÁ¦×÷ÓõÄʱ¼äÊDz»Í¬µÄ£»¶øÔÚ¹ý³ÌµÄ³õ̬ºÍĩ̬,ÈËÌåµÄËٶȾùΪÁ㣮ÕâÑù,ÔËÓö¯Á¿¶¨ÀíÈԿɵõ½ÏàͬµÄ½á¹û£®
½â1 ÒÔÈËΪÑо¿¶ÔÏó,°´·ÖÎöÖеÄÁ½¸ö½×¶Î½øÐÐÌÖÂÛ£®ÔÚ×ÔÓÉÂäÌåÔ˶¯¹ý³ÌÖÐ,È˵øÂäÖÁ2 m ´¦Ê±µÄËÙ¶ÈΪ
v1?2gh (1)
ÔÚ»º³å¹ý³ÌÖÐ,ÈËÊÜÖØÁ¦ºÍ°²È«´ø³åÁ¦µÄ×÷ÓÃ,¸ù¾Ý¶¯Á¿¶¨Àí,ÓÐ
?F?P?¦¤t?mv2?mv1 (2)
ÓÉʽ(1)¡¢(2)¿ÉµÃ°²È«´ø¶ÔÈ˵Äƽ¾ù³åÁ¦´óСΪ
F?mg?¦¤2gh¦¤?mv??mg??1.14?103N
¦¤t¦¤t½â2 ´ÓÕû¸ö¹ý³ÌÀ´ÌÖÂÛ£®¸ù¾Ý¶¯Á¿¶¨ÀíÓÐ
F?mg2h/g?mg?1.14?103N ¦¤t3 -10 ÖÊÁ¿Îªm µÄСÇò,ÔÚºÏÍâÁ¦F £½-kx ×÷ÓÃÏÂÔ˶¯,ÒÑÖªx £½Acos¦Øt,ÆäÖÐk¡¢¦Ø¡¢A ¾ùΪÕý³£Á¿,ÇóÔÚ
t £½0 µ½t?¦Ð ʱ¼äÄÚСÇò¶¯Á¿µÄÔöÁ¿£® 2¦Ø·ÖÎö ÓɳåÁ¿¶¨ÒåÇóµÃÁ¦FµÄ³åÁ¿ºó,¸ù¾Ý¶¯Á¿ÔÀí,¼´Îª¶¯Á¿ÔöÁ¿,×¢ÒâÓÃʽÓÃx £½Acos¦Øt´úÖ®,·½ÄÜ»ý·Ö£® ½â Á¦F µÄ³åÁ¿Îª
?t2t1Fdt»ý·ÖÇ°,Ó¦ÏȽ«Ê½ÖÐx
I??t1Fdt??t1?kxdt???0¼´ ¦¤3 -11 Ò»Ö»ÖÊÁ¿m1t2t2¦Ð/2?kAcos?tdt??kA?
?mv???kA
¦Ø?0.11kgµÄÀÝÇòÒÔv1?17m?s?1ˮƽËÙÂÊÈÓÏò´ò»÷ÊÖ£¬Çò¾Çò°ô»÷³öºó£¬¾ß
ÓÐÈçͼ£¨a£©ËùʾµÄËÙ¶ÈÇÒ´óСv2?34m?s?1£¬ÈôÇòÓë°ôµÄ½Ó´¥Ê±¼äΪ0.025 s,Ç󣺣¨1£©°ô¶Ô¸ÃÇòƽ
¾ù×÷ÓÃÁ¦µÄ´óС£»£¨2£©ÀÝÇòÊÖÖÁÉÙ¶ÔÇò×÷Á˶àÉÙ¹¦£¿
·ÖÎö µÚ£¨1£©ÎʿɶÔÀÝÇòÔËÓö¯Á¿¶¨Àí£¬¼È¿É¸ù¾Ý¶¯Á¿¶¨ÀíµÄʸÁ¿Ê½£¬Óü¸ºÎ·¨Çó½â£¬Èçͼ£¨b£©Ëùʾ£»
32
Ò²¿É½¨Á¢Èçͼ£¨a£©ËùʾµÄ×ø±êϵ£¬Óö¯Á¿¶¨Á¿µÄ·ÖÁ¿Ê½Çó½â£¬¶Ô´ò»÷¡¢ÅöײһÀà×÷ÓÃʱ¼äºÜ¶ÌµÄ¹ý³ÌÀ´Ëµ£¬ÎïÌåµÄÖØÁ¦Ò»°ã¿ÉÂÔÈ¥²»¼Æ.
Ìâ 3-11 ͼ
½â £¨1£© ½â 1 ÓÉ·ÖÎöÖª£¬ÓÐ
F?t?mv2?mv1
ÆäʸÁ¿¹ØϵÈçͼ£¨b£©Ëùʾ£¬Ôò
(F?t)2?(mv1)2?(mv2)2?2(mv1)(mv2)cos(180??60?)
½âÖ®µÃ F½â 2 ÓÉͼ£¨a£©ÓÐ
?197.9N
Fx?t?mv2x?mv1x Fy?t?mv2y?0
½«v1x?v,v2x??v2cos60?¼°v2y?v2sin60?´úÈë½âµÃFxºÍFy£¬Ôò
F?F?F?197.9N
(2) ÓÉÖʵ㶯Äܶ¨Àí£¬µÃ
2x2y1212W?mv2?mv1?47.7J
223 -12 ÈçͼËùʾ,ÔÚˮƽµØÃæÉÏ,ÓÐÒ»ºá½ØÃæS £½0.20 m µÄÖ±½ÇÍä¹Ü,¹ÜÖÐÓÐÁ÷ËÙΪv £½3.0 m¡¤£ó µÄˮͨ¹ý,ÇóÍä¹ÜËùÊÜÁ¦µÄ´óСºÍ·½Ïò£®
2
-1
33
Ìâ 3-12 ͼ
·ÖÎö ¶ÔÓÚÍäÇú²¿·ÖAB ¶ÎÄÚµÄË®¶øÑÔ,ÓÉÓÚÁ÷ËÙÒ»¶¨,ÔÚʱ¼ä¦¤t ÄÚ,´ÓÆäÒ»¶ËÁ÷ÈëµÄË®Á¿µÈÓÚ´ÓÁíÒ»¶ËÁ÷³öµÄË®Á¿£®Òò´Ë,¶ÔÕⲿ·ÖË®À´Ëµ,ÔÚʱ¼ä¦¤t ÄÚ¶¯Á¿µÄÔöÁ¿Ò²¾ÍÊÇÁ÷ÈëÓëÁ÷³öË®µÄ¶¯Á¿µÄÔöÁ¿¦¤p£½¦¤m(vB -vA )£»´Ë¶¯Á¿µÄ±ä»¯ÊǹܱÚÔÚ¦¤tʱ¼äÄÚ¶ÔÆä×÷ÓóåÁ¿I µÄ½á¹û£®ÒÀ¾Ý¶¯Á¿¶¨Àí¿ÉÇóµÃ¸Ã¶ÎË®Êܵ½¹Ü±ÚµÄ³åÁ¦F£»ÓÉÅ£¶ÙµÚÈý¶¨ÂÉ,×ÔÈ»¾ÍµÃµ½Ë®Á÷¶Ô¹Ü±ÚµÄ×÷ÓÃÁ¦F¡ä£½-F£®
½â ÔÚ¦¤t ʱ¼äÄÚ,´Ó¹ÜÒ»¶ËÁ÷Èë(»òÁ÷³ö) Ë®µÄÖÊÁ¿Îª¦¤m £½¦Ñ¦ÔS¦¤t,ÍäÇú²¿·ÖAB µÄË®µÄ¶¯Á¿µÄÔöÁ¿ÔòΪ
¦¤p£½¦¤m(vB -vA ) £½¦Ñ¦ÔS¦¤t (vB -vA )
ÒÀ¾Ý¶¯Á¿¶¨ÀíI £½¦¤p,µÃµ½¹Ü±Ú¶ÔÕⲿ·ÖË®µÄƽ¾ù³åÁ¦
F?´Ó¶ø¿ÉµÃË®Á÷¶Ô¹Ü±Ú×÷ÓÃÁ¦µÄ´óСΪ
I?¦ÑSv¦¤t?vB?vA? ¦¤tF???F??2¦ÑSv2??2.5?103N
×÷ÓÃÁ¦µÄ·½ÏòÔòÑØÖ±½Çƽ·ÖÏßÖ¸ÏòÍä¹ÜÍâ²à£®
3 -13 A¡¢B Á½´¬ÔÚƽ¾²µÄºþÃæÉÏƽÐÐÄæÏòº½ÐÐ,µ±Á½´¬²Á¼çÏàÓöʱ,Á½´¬¸÷×ÔÏò¶Ô·½Æ½Îȵش«µÝ50 kg µÄÖØÎï,½á¹ûÊÇA ´¬Í£ÁËÏÂÀ´,¶øB ´¬ÒÔ3.4 m¡¤sµÄËٶȼÌÐøÏòǰʻȥ£®A¡¢B Á½´¬ÔÓÐÖÊÁ¿·Ö±ðΪ0.5¡Á10 kg ºÍ1.0 ¡Á10 kg,ÇóÔÚ´«µÝÖØÎïÇ°Á½´¬µÄËٶȣ®(ºöÂÔË®¶Ô´¬µÄ×èÁ¦)
·ÖÎö ÓÉÓÚÁ½´¬ºáÏò´«µÝµÄËٶȿÉÂÔÈ¥²»¼Æ,Ôò¶Ô°á³öÖØÎïºóµÄ´¬AÓë´Ó´¬B°áÈëµÄÖØÎïËù×é³ÉµÄϵͳ¢ñÀ´½²,ÔÚˮƽ·½ÏòÉÏÎÞÍâÁ¦×÷ÓÃ,Òò´Ë,ËüÃÇÏ໥×÷ÓõĹý³ÌÖÐÓ¦Âú×㶯Á¿Êغ㣻ͬÑù,¶Ô°á³öÖØÎïºóµÄ´¬BÓë´Ó´¬A°áÈëµÄÖØÎïËù×é³ÉµÄϵͳ¢òÒàÊÇÕâÑù£®ÓÉ´Ë,·Ö±ðÁгöϵͳ¢ñ¡¢¢òµÄ¶¯Á¿Êغ㷽³Ì¼´¿É½â³ö½á¹û£® ½â ÉèA¡¢BÁ½´¬ÔÓеÄËٶȷֱðÒÔvA ¡¢vB ±íʾ,´«µÝÖØÎïºó´¬µÄËٶȷֱðÒÔvA¡ä ¡¢vB¡ä ±íʾ,±»°áÔËÖØÎïµÄÖÊÁ¿ÒÔm ±íʾ£®·Ö±ð¶ÔÉÏÊöϵͳ¢ñ¡¢¢òÓ¦Óö¯Á¿Êغ㶨ÂÉ,ÔòÓÐ
3
-1
3
?mB?m?vB?mvA?mBv?B? (2)
ÓÉÌâÒâÖªvA¡ä £½0, vB¡ä £½3.4 m¡¤s ´úÈëÊý¾Ýºó,¿É½âµÃ
-1
?mA?m?vA?mvB?mAv?A (1)
vA??mBmv?B??0.40m?s?1 2?mB?m??mA?m??mvB??mA?m?mBv?B?mA?m??mB?m??m234
?3.6m?s?1
Ò²¿ÉÒÔÑ¡Ôñ²»Í¬µÄϵͳ,ÀýÈç,°ÑA¡¢BÁ½´¬(°üÀ¨´«µÝµÄÎïÌåÔÚÄÚ)ÊÓΪϵͳ,ͬÑùÄÜÂú×㶯Á¿Êغã,Ò²¿ÉÁгöÏà¶ÔÓ¦µÄ·½³ÌÇó½â£®
3 -14 ÖÊÁ¿Îªm¡ä µÄÈËÊÖÀïÄÃ×ÅÒ»¸öÖÊÁ¿Îªm µÄÎïÌå,´ËÈËÓÃÓëˮƽÃæ³É¦Á½ÇµÄËÙÂÊv0 ÏòÇ°ÌøÈ¥£®µ±Ëû´ïµ½×î¸ßµãʱ,Ëû½«ÎïÌåÒÔÏà¶ÔÓÚÈËΪu µÄˮƽËÙÂÊÏòºóÅ׳ö£®ÎÊ£ºÓÉÓÚÈËÅ׳öÎïÌå,ËûÌøÔ¾µÄ¾àÀëÔö¼ÓÁ˶àÉÙ£¿ (¼ÙÉèÈË¿ÉÊÓΪÖʵã)
·ÖÎö ÈËÌøÔ¾¾àÀëµÄÔö¼ÓÊÇÓÉÓÚËûÔÚ×î¸ßµã´¦ÏòºóÅ׳öÎïÌåËùÖ£®ÔÚÅ×ÎïµÄ¹ý³ÌÖÐ,ÈËÓëÎïÖ®¼äÏ໥×÷ÓÃÁ¦µÄ³åÁ¿,ʹËûÃǸ÷×ԵĶ¯Á¿·¢ÉúÁ˱仯£®Èç¹û°ÑÈËÓëÎïÊÓΪһϵͳ,Òòˮƽ·½Ïò²»ÊÜÍâÁ¦×÷ÓÃ,¹ÊÍâÁ¦µÄ³åÁ¿ÎªÁã,ϵͳÔڸ÷½ÏòÉ϶¯Á¿Êغ㣮µ«ÔÚÓ¦Óö¯Á¿Êغ㶨ÂÉʱ,±ØÐë×¢ÒâϵͳÊÇÏà¶ÔµØÃæ(¹ßÐÔϵ)¶øÑÔµÄ,Òò´Ë,ÔÚ´¦ÀíÈËÓëÎïµÄËÙ¶Èʱ,Òª¸ù¾ÝÏà¶ÔÔ˶¯µÄ¹ØϵÀ´È·¶¨£®ÖÁÓÚ,ÈËÒòÌøÔ¾¶øÔö¼ÓµÄ¾àÀë,¿É¸ù¾ÝÈËÔÚˮƽ·½ÏòËÙÂʵÄÔöÁ¿¦¤v À´¼ÆË㣮
½â È¡ÈçͼËùʾ×ø±ê£®°ÑÈËÓëÎïÊÓΪһϵͳ,µ±ÈËÌøÔ¾µ½×î¸ßµã´¦,ÔÚÏò×óÅ×ÎïµÄ¹ý³ÌÖÐ,Âú×㶯Á¿Êغã,¹ÊÓÐ
?m?m??v0cos¦Á?m?v?m?v?u?
ʽÖÐv ΪÈËÅ×ÎïºóÏà¶ÔµØÃæµÄˮƽËÙÂÊ, v -u ΪÅ׳öÎï¶ÔµØÃæµÄˮƽËÙÂÊ£®µÃ
v0?v0cos¦Á?È˵ÄˮƽËÙÂʵÄÔöÁ¿Îª
mu
m?m?mu ?m?m¦¤v?v?v0cos¦Á?¶øÈË´Ó×î¸ßµãµ½µØÃæµÄÔ˶¯Ê±¼äΪ
t?ËùÒÔ,ÈËÌøÔ¾ºóÔö¼ÓµÄ¾àÀë
v0sin¦Á g¦¤x?¦¤vt?3
mv0sin¦Á
?m?m??g3 -15 Ò»ÎïÌåÔÚ½éÖÊÖа´¹æÂÉx £½ct ×÷Ö±ÏßÔ˶¯,c Ϊһ³£Á¿£®Éè½éÖʶÔÎïÌåµÄ×èÁ¦Õý±ÈÓÚËٶȵÄƽ·½£®ÊÔÇóÎïÌåÓÉx0 £½0 Ô˶¯µ½x £½l ʱ,×èÁ¦Ëù×÷µÄ¹¦£®(ÒÑÖª×èÁ¦ÏµÊýΪk) ·ÖÎö ±¾ÌâÊÇһά±äÁ¦×÷¹¦ÎÊÌâ,ÈÔÐè°´¹¦µÄ¶¨ÒåʽW??F?dxÀ´Çó½â£®¹Ø¼üÔÚÓÚÑ°ÕÒÁ¦º¯ÊýF £½
F(x)£®¸ù¾ÝÔ˶¯Ñ§¹Øϵ,¿É½«ÒÑÖªÁ¦ÓëËٶȵĺ¯Êý¹ØϵF(v) £½kv2 ±ä»»µ½F(t),½øÒ»²½°´x £½ct3 µÄ¹Øϵ°Ñ
35
F(t)ת»»ÎªF(x),ÕâÑù,¾Í¿É°´¹¦µÄ¶¨ÒåʽÇó½â£®
½â ÓÉÔ˶¯Ñ§·½³Ìx £½ct ,¿ÉµÃÎïÌåµÄËÙ¶È
3
v?°´ÌâÒâ¼°ÉÏÊö¹Øϵ,ÎïÌåËùÊÜ×èÁ¦µÄ´óСΪ
dx?3ct2 dtF?kv2?9kc2t4?9kc2/3x4/3
Ôò×èÁ¦µÄ¹¦Îª
?2/34/3W??F?dxW??0F?dx??cos180dx???09kcxdx??ll272/37/3kcl 73 -16 Ò»ÈË´Ó10.0 m ÉîµÄ¾®ÖÐÌáË®,ÆðʼͰÖÐ×°ÓÐ10.0 kg µÄË®,ÓÉÓÚˮͰ©ˮ,ÿÉý¸ß1.00 m Ҫ©ȥ0.20 kg µÄË®£®Ë®Í°±»ÔÈËٵشӾ®ÖÐÌáµ½¾®¿Ú,ÇóËù×÷µÄ¹¦£®
Ìâ 3-16 ͼ
·ÖÎö ÓÉÓÚˮͰÔÚÔÈËÙÉÏÌá¹ý³ÌÖÐ,ÀÁ¦±ØÐëʼÖÕÓëˮͰÖØÁ¦Ïàƽºâ£®Ë®Í°ÖØÁ¦Òò©ˮ¶øËæÌáÉý¸ß¶È¶ø±ä,Òò´Ë,ÀÁ¦×÷¹¦ÊµÎª±äÁ¦×÷¹¦£®ÓÉÓÚÀÁ¦×÷¹¦Ò²¾ÍÊÇ¿Ë·þÖØÁ¦µÄ¹¦,Òò´Ë,Ö»ÒªÄÜд³öÖØÁ¦Ëæ¸ß¶È±ä»¯µÄ¹Øϵ,ÀÁ¦×÷¹¦¼´¿ÉÇó³ö£®
½â ˮͰÔÚÔÈËÙÉÏÌá¹ý³ÌÖÐ,a £½0,ÀÁ¦ÓëˮͰÖØÁ¦Æ½ºâ,ÓÐ
F £«P £½0
ÔÚͼʾËùÈ¡×ø±êÏÂ,ˮͰÖØÁ¦ËæλÖõı仯¹ØϵΪ
P £½mg -¦Ágy
ÆäÖЦÁ£½0.2 kg/m,È˶ÔˮͰµÄÀÁ¦µÄ¹¦Îª
W??0F?dy??0?mg?agy?dy?882J
3 -17 Ò»ÖÊÁ¿Îª0.20 kg µÄÇò,ϵÔÚ³¤Îª2.00 m µÄϸÉþÉÏ,ϸÉþµÄÁíÒ»¶ËϵÔÚÌ컨°åÉÏ£®°ÑСÇòÒÆÖÁʹϸÉþÓëÊúÖ±·½Ïò³É30¡ã½ÇµÄλÖÃ,È»ºó´Ó¾²Ö¹·Å¿ª£®Çó£º(1) ÔÚÉþË÷´Ó30¡ã½Çµ½0¡ã½ÇµÄ¹ý³ÌÖÐ,ÖØÁ¦ºÍÕÅÁ¦Ëù×÷µÄ¹¦£»(2) ÎïÌåÔÚ×îµÍλÖÃʱµÄ¶¯ÄܺÍËÙÂÊ£»(3) ÔÚ×îµÍλÖÃʱµÄÕÅÁ¦£®
1010 36
Ìâ 3-17 ͼ
·ÖÎö (1) ÔÚ¼ÆË㹦ʱ,Ê×ÏÈÓ¦Ã÷È·ÊÇʲôÁ¦×÷¹¦£®Ð¡Çò°Ú¶¯¹ý³ÌÖÐͬʱÊܵ½ÖØÁ¦ºÍÕÅÁ¦×÷Óã®ÖØÁ¦ÊDZ£ÊØÁ¦,¸ù¾ÝСÇòÏÂÂäµÄ¾àÀë,ËüµÄ¹¦ºÜÒ×ÇóµÃ£»ÖÁÓÚÕÅÁ¦ËäÊÇÒ»±äÁ¦,µ«ÊÇ,ËüµÄ·½ÏòʼÖÕÓëСÇòÔ˶¯·½Ïò´¹Ö±,¸ù¾Ý¹¦µÄʸÁ¿Ê½W??F?ds,¼´Äܵóö½á¹ûÀ´£®(2) ÔÚ¼ÆË㹦µÄ»ù´¡ÉÏ,Óɶ¯Äܶ¨ÀíÖ±½ÓÄÜÇó³ö¶¯ÄܺÍ
ËÙÂÊ£®(3) ÔÚÇó×îµÍµãµÄÕÅÁ¦Ê±,¿É¸ù¾ÝСÇò×÷Ô²ÖÜÔ˶¯Ê±µÄÏòÐļÓËÙ¶ÈÓÉÖØÁ¦ºÍÕÅÁ¦ÌṩÀ´È·¶¨£® ½â (1) ÈçͼËùʾ,ÖØÁ¦¶ÔСÇòËù×÷µÄ¹¦Ö»ÓëʼĩλÖÃÓйØ,¼´
WP?P¦¤h?mgl?1?cos¦È??0.53J
ÔÚСÇò°Ú¶¯¹ý³ÌÖÐ,ÕÅÁ¦F£Ô µÄ·½Ïò×ÜÊÇÓëÔ˶¯·½Ïò´¹Ö±,ËùÒÔ,ÕÅÁ¦µÄ¹¦
WT??FT?ds
(2) ¸ù¾Ý¶¯Äܶ¨Àí,СÇò°Ú¶¯¹ý³ÌÖÐ,Æ䶯ÄܵÄÔöÁ¿ÊÇÓÉÓÚÖØÁ¦¶ÔËü×÷¹¦µÄ½á¹û£®³õʼʱ¶¯ÄÜΪÁã,Òò¶ø,ÔÚ×îµÍλÖÃʱµÄ¶¯ÄÜΪ
Ek?WP?0.53J
СÇòÔÚ×îµÍλÖõÄËÙÂÊΪ
v?2EK2WP??2.30m?s?1 mm(3) µ±Ð¡ÇòÔÚ×îµÍλÖÃʱ,ÓÉÅ£¶Ù¶¨ÂɿɵÃ
mv2FT?P?l
mv2FT?mg??2.49N
l3 -18 Ò»ÖÊÁ¿Îªm µÄÖʵã,ϵÔÚϸÉþµÄÒ»¶Ë,ÉþµÄÁíÒ»¶Ë¹Ì¶¨ÔÚƽÃæÉÏ£®´ËÖʵãÔÚ´Ö²ÚˮƽÃæÉÏ×÷°ë¾¶Îªr µÄÔ²ÖÜÔ˶¯£®ÉèÖʵãµÄ×î³õËÙÂÊÊÇv0 £®µ±ËüÔ˶¯Ò»ÖÜʱ,ÆäËÙÂÊΪv0 /2£®Çó£º(1) Ħ²ÁÁ¦×÷µÄ¹¦£»(2) ¶¯Ä¦²ÁÒòÊý£»(3) ÔÚ¾²Ö¹ÒÔÇ°ÖʵãÔ˶¯Á˶àÉÙȦ£¿
·ÖÎö ÖʵãÔÚÔ˶¯¹ý³ÌÖÐËٶȵļõ»º,Òâζ×ÅÆ䶯ÄܼõÉÙ£»¶ø¼õÉÙµÄÕⲿ·Ö¶¯ÄÜÔòÏûºÄÔÚÔ˶¯Öп˷þĦ²ÁÁ¦×÷¹¦ÉÏ£®ÓÉ´Ë,¿ÉÒÀ¾Ý¶¯Äܶ¨ÀíÁÐʽ½âÖ®£®
37
½â (1) Ħ²ÁÁ¦×÷¹¦Îª
W11232 (1) ?Ek?Ek0?mv2?mv0??mv0228(2) ÓÉÓÚĦ²ÁÁ¦ÊÇÒ»ºãÁ¦,ÇÒF£æ £½¦Ìmg,¹ÊÓÐ
W?Ffscos180o??2?r?mg (2)
ÓÉʽ(1)¡¢(2)¿ÉµÃ¶¯Ä¦²ÁÒòÊýΪ
23v0 ¦Ì?16¦Ðrg(3) ÓÉÓÚÒ»ÖÜÖÐËðʧµÄ¶¯ÄÜΪ
32
mv0,ÔòÔÚ¾²Ö¹Ç°¿ÉÔËÐеÄȦÊýΪ 8
n?Ek04?Ȧ W33 -19 Èçͼ(a)Ëùʾ,A ºÍB Á½¿é°åÓÃÒ»Çᵯ»ÉÁ¬½ÓÆðÀ´,ËüÃǵÄÖÊÁ¿·Ö±ðΪm1 ºÍm2 £®ÎÊÔÚA °åÉÏÐè¼Ó¶à´óµÄѹÁ¦,·½¿ÉÔÚÁ¦Í£Ö¹×÷Óúó,Ç¡ÄÜʹA ÔÚÌøÆðÀ´Ê±B ÉÔ±»ÌáÆð£®(É赯»ÉµÄ¾¢¶ÈϵÊýΪk)
Ìâ 3-19 ͼ
·ÖÎö ÔËÓÃÊغ㶨ÂÉÇó½âÊǽâ¾öÁ¦Ñ§ÎÊÌâ×î¼ò½ÝµÄ;¾¶Ö®Ò»£®ÒòΪËüÓë¹ý³ÌµÄϸ½ÚÎÞ¹Ø,Ò²³£³£ÓëÌض¨Á¦µÄϸ½ÚÎ޹أ®¡°Êغ㡱ÔòÒâζ×ÅÔÚÌõ¼þÂú×ãµÄÇ°ÌáÏÂ,¹ý³ÌÖÐÈκÎʱ¿ÌÊغãÁ¿²»±ä£®ÔÚ¾ßÌåÓ¦ÓÃʱ,±ØÐëÇ¡µ±µØÑ¡È¡Ñо¿¶ÔÏó(ϵͳ),×¢ÒâÊغ㶨ÂɳÉÁ¢µÄÌõ¼þ£®¸ÃÌâ¿ÉÓûúеÄÜÊغ㶨ÂÉÀ´½â¾ö£®Ñ¡È¡Á½¿é°å¡¢µ¯»ÉºÍµØÇòΪϵͳ,¸ÃϵͳÔÚÍâ½çËùʩѹÁ¦³·³ýºó(È¡×÷״̬1),Ö±µ½B °å¸Õ±»ÌáÆð(È¡×÷״̬2),ÔÚÕâÒ»¹ý³ÌÖÐ,ϵͳ²»ÊÜÍâÁ¦×÷ÓÃ,¶øÄÚÁ¦ÖÐÓÖÖ»Óб£ÊØÁ¦(ÖØÁ¦ºÍµ¯Á¦)×÷¹¦,Ö§³ÖÁ¦²»×÷¹¦,Òò´Ë,Âú×ã»úеÄÜÊغãµÄÌõ¼þ£®Ö»Ðèȡ״̬1 ºÍ״̬2,ÔËÓûúеÄÜÊغ㶨ÂÉÁгö·½³Ì,²¢½áºÏÕâÁ½×´Ì¬ÏÂÊÜÁ¦µÄƽºâ,±ã¿É½«ËùÐèѹÁ¦Çó³ö£®
½â Ñ¡È¡Èçͼ(b)Ëùʾ×ø±ê,È¡ÔµãO´¦ÎªÖØÁ¦ÊÆÄܺ͵¯ÐÔÊÆÄÜÁãµã£®×÷¸÷״̬ÏÂÎïÌåµÄÊÜÁ¦Í¼£®¶ÔA °å¶øÑÔ,µ±Ê©ÒÔÍâÁ¦F ʱ,¸ù¾ÝÊÜÁ¦Æ½ºâÓÐ
F1 £½P1 £«F (1)
µ±ÍâÁ¦³·³ýºó,°´·ÖÎöÖÐËùÑ¡µÄϵͳ,ÓÉ»úеÄÜÊغ㶨ÂɿɵÃ
1212ky1?mgy1?ky2?mgy2 22ʽÖÐy1 ¡¢y2 ΪM¡¢N Á½µã¶ÔÔµãO µÄλÒÆ£®ÒòΪF1 £½ky1 ,F2 £½ky2 ¼°P1 £½m1g,ÉÏʽ¿ÉдΪ
F1 -F2 £½2P1 (2)
38
ÓÉʽ(1)¡¢(2)¿ÉµÃ
F £½P1 £«F2 (3)
µ±A °åÌøµ½N µãʱ,B °å¸Õ±»ÌáÆð,´Ëʱµ¯ÐÔÁ¦F¡ä2 £½P2 ,ÇÒF2 £½F¡ä2 £®ÓÉʽ(3)¿ÉµÃ
F £½P1 £«P2 £½(m1 £«m2 )g
ӦעÒâ,ÊÆÄܵÄÁãµãλÖÃÊÇ¿ÉÒÔÈÎÒâÑ¡È¡µÄ£®Îª¼ÆËã·½±ãÆð¼û,ͨ³£È¡µ¯»ÉÔ³¤Ê±µÄµ¯ÐÔÊÆÄÜΪÁãµã,ҲͬʱΪÖØÁ¦ÊÆÄܵÄÁãµã£®
3 -20 ÈçͼËùʾ£¬Ò»ÖÊÁ¿ÎªmµÄľ¿é¾²Ö¹Ôڹ⻬ˮƽÃæÉÏ£¬Ò»ÖÊÁ¿Îªm/2µÄ×Óµ¯ÑØˮƽ·½ÏòÒÔËÙÂÊv0ÉäÈëľ¿éÒ»¶Î¾àÀëL(´Ëʱľ¿é»¬ÐоàÀëǡΪs)ºóÁôÔÚľ¿éÄÚ£¬Ç󣺣¨1£©Ä¾¿éÓë×Óµ¯µÄ¹²Í¬ËÙ¶Èv,´Ë¹ý³ÌÖÐľ¿éºÍ×Óµ¯µÄ¶¯Äܸ÷±ä»¯Á˶àÉÙ£¿£¨2£©×Óµ¯Óëľ¿é¼äµÄĦ²Á×èÁ¦¶Ôľ¿éºÍ×Óµ¯¸÷×÷Á˶àÉÙ¹¦£¿£¨3£©Ö¤Ã÷ÕâÒ»¶ÔĦ²Á×èÁ¦µÄËù×÷¹¦µÄ´úÊýºÍ¾ÍµÈÓÚÆäÖÐÒ»¸öĦ²Á×èÁ¦ÑØÏà¶ÔλÒÆLËù×÷µÄ¹¦.(4)Ö¤Ã÷ÕâÒ»¶ÔĦ²Á×èÁ¦Ëù×÷¹¦µÄ´úÊýºÍ¾ÍµÈÓÚ×Óµ¯-ľ¿éϵͳ×Ü»úеÄܵļõÉÙÁ¿(Ò༴ת»¯ÎªÈȵÄÄDz¿·ÖÄÜÁ¿).
Ìâ 3-20 ͼ
·ÖÎö ¶Ô×Óµ¯-ľ¿éϵͳÀ´Ëµ£¬Âú×㶯Á¿Êغ㣬µ«ÏµÍ³¶¯Äܲ¢²»Êغ㣬ÕâÊÇÒòΪһ¶ÔĦ²ÁÄÚÁ¦Ëù×ö¹¦µÄ´úÊýºÍ²¢²»ÎªÁ㣬ÆäÖÐĦ²Á×èÁ¦¶Ôľ¿é×÷Õý¹¦£¬Æä·´×÷ÓÃÁ¦¶Ô×Óµ¯×÷¸º¹¦£¬ºóÕß¹¦µÄÊýÖµ´óÓÚÇ°Õߣ¬Í¨¹ýÕâÒ»¶Ô×÷ÓÃÁ¦Óë·´×÷ÓÃÁ¦Ëù×ö¹¦£¬×Óµ¯½«Ò»²¿·Ö¶¯ÄÜתÒƸøľ¿é£¬¶øÁíÒ»²¿·Öȴת»¯ÎªÎïÌåÄÚÄÜ.±¾Ì⣨3£©¡¢£¨4£©Á½Îʸø³öÁ˾ßÓÐÆÕ±éÒâÒåµÄ½áÂÛ£¬¿É°ïÖú¶ÁÕßÒÔºó·ÖÎö´ËÀàÎÊÌâ. ½â £¨1£©×Óµ¯-ľ¿éϵͳÂú×㶯Á¿Êغ㣬ÓÐ
mv0/2?(m/2?m)v
½âµÃ¹²Í¬ËÙ¶È
1v?v0
3¶Ôľ¿é ?Ek2112?mv2?0?mv0 218¶Ô×Óµ¯ ?Ek21m1m222?()v2?()v0??mv0 2222912mv0 18£¨2£© ¶Ôľ¿éºÍ×Óµ¯·Ö±ðÔËÓÃÖʵ㶯Äܶ¨Àí£¬Ôò ¶Ôľ¿é W1??Ek1?¶Ô×Óµ¯ W222??Ek2??mv0
9 £¨3£© ÉèĦ²Á×èÁ¦´óСΪFf,ÔÚÁ½ÕßÈ¡µÃ¹²Í¬ËÙ¶Èʱ£¬Ä¾¿é¶ÔµØλÒÆΪs£¬Ôò×Óµ¯¶ÔµØλÒÆΪL+s,ÓÐ
39
¶Ôľ¿é W1¶Ô×Óµ¯ W2µÃ W?Ffs
??Ff(L?s)
?W1?W2??FfL
ʽÖÐL¼´Îª×Óµ¯¶Ôľ¿éµÄÏà¶ÔλÒÆ£¬¡°-¡±ºÅ±íʾÕâÒ»¶ÔĦ²Á×èÁ¦£¨·Ç±£ÊØÁ¦£©Ëù×÷¹¦±Ø¶¨»áʹϵͳ»úеÄܼõÉÙ.
(4) ¶Ôľ¿é W11?Ffs?mv2
2¶Ô×Óµ¯ W2Á½Ê½Ïà¼Ó£¬µÃ W11m1m2??Ff(L?s)?()v2?()v0
222211m1m2?W2?[mv2?()v2]?()v0
22222FfL??32mv0 18¼´ ?Á½Ê½Ïà¼ÓºóʵΪ×Óµ¯-ľ¿éϵͳ×÷ΪÖʵãϵµÄ¶¯Äܶ¨Àí±í´ïʽ£¬×ó±ßΪһ¶ÔÄÚÁ¦Ëù×÷¹¦£¬ÓÒ±ßΪϵͳ¶¯Äܵı仯Á¿.
3 -21 ÓÃÌú´¸°Ñ¶¤×ÓÇÃÈëǽÃæľ°å£®Éèľ°å¶Ô¶¤×ÓµÄ×èÁ¦Ó붤×Ó½øÈëľ°åµÄÉî¶È³ÉÕý±È£®ÈôµÚÒ»´ÎÇû÷,ÄܰѶ¤×Ó¶¤Èëľ°å1.00 ¡Á10 m£®µÚ¶þ´ÎÇû÷ʱ,±£³ÖµÚÒ»´ÎÇû÷¶¤×ÓµÄËÙ¶È,ÄÇôµÚ¶þ´ÎÄܰѶ¤×Ó¶¤Èë¶àÉ
·ÖÎö ÓÉÓÚÁ½´Î´¸»÷µÄÌõ¼þÏàͬ,´¸»÷ºó¶¤×Ó»ñµÃµÄËÙ¶ÈÒ²Ïàͬ,Ëù¾ßÓеijõ¶¯ÄÜÒ²Ïàͬ£®¶¤×Ó¶¤Èëľ°åÊǽ«¶¤×ӵĶ¯ÄÜÓÃÓÚ¿Ë·þ×èÁ¦×÷¹¦,Óɹ¦ÄÜÔÀí¿ÉÖª¶¤×ÓÁ½´ÎËù×÷µÄ¹¦ÏàµÈ£®ÓÉÓÚ×èÁ¦Óë½øÈëľ°åµÄÉî¶È³ÉÕý±È,°´±äÁ¦µÄ¹¦µÄ¶¨ÒåµÃÁ½´Î¹¦µÄ±í´ïʽ,²¢Óɹ¦ÏàµÈµÄ¹Øϵ¼´¿ÉÇó½â£®
½â Òò×èÁ¦ÓëÉî¶È³ÉÕý±È,ÔòÓÐF£½kx(k Ϊ×èÁ¦ÏµÊý)£®ÏÖÁîx0£½1.00 ¡Á10 m,µÚ¶þ´Î¶¤ÈëµÄÉî¶ÈΪ¦¤x,ÓÉÓÚ¶¤×ÓÁ½´ÎËù×÷¹¦ÏàµÈ,¿ÉµÃ
-2
-2
?x00kxdx??x0?¦¤xx0kxdx
-2
¦¤x£½0.41 ¡Á10 m
3 -22 Ò»ÖÊÁ¿Îªm µÄµØÇòÎÀÐÇ,Ñذ뾶Ϊ3RE µÄÔ²¹ìµÀÔ˶¯, REΪµØÇòµÄ°ë¾¶£®ÒÑÖªµØÇòµÄÖÊÁ¿ÎªmE£®Çó£º(1) ÎÀÐǵĶ¯ÄÜ£»(2) ÎÀÐǵÄÒýÁ¦ÊÆÄÜ£»(3) ÎÀÐǵĻúеÄÜ£®
·ÖÎö ¸ù¾ÝÊÆÄܺͶ¯ÄܵĶ¨Òå,Ö»ÐèÖªµÀÎÀÐǵÄËùÔÚλÖúÍÈƵØÇòÔ˶¯µÄËÙÂÊ,ÆäÊÆÄܺͶ¯Äܼ´¿ÉËã³ö£®ÓÉÓÚÎÀÐÇÔÚµØÇòÒýÁ¦×÷ÓÃÏÂ×÷Ô²ÖÜÔ˶¯,ÓÉ´Ë¿ÉËãµÃÎÀÐÇÈƵØÇòÔ˶¯µÄËÙÂʺͶ¯ÄÜ£®ÓÉÓÚÎÀÐǵÄÒýÁ¦ÊÆÄÜÊÇÊôÓÚϵͳ(ÎÀÐǺ͵ØÇò)µÄ,Ҫȷ¶¨Ìض¨Î»ÖõÄÊÆÄÜʱ,±ØÐë¹æ¶¨ÊÆÄܵÄÁãµã,ͨ³£È¡ÎÀÐÇÓëµØÇòÏà¾àÎÞÏÞԶʱµÄÊÆÄÜΪÁ㣮ÕâÑù,ÎÀÐÇÔÚÌض¨Î»ÖõÄÊÆÄÜÒ²¾ÍÄÜÈ·¶¨ÁË£®ÖÁÓÚÎÀÐǵĻúеÄÜÔòÊǶ¯ÄܺÍÊÆÄܵÄ×ܺͣ®
40
½â (1) ÎÀÐÇÓëµØÇòÖ®¼äµÄÍòÓÐÒýÁ¦ÌṩÎÀÐÇ×÷Ô²ÖÜÔ˶¯µÄÏòÐÄÁ¦,ÓÉÅ£¶Ù¶¨ÂɿɵÃ
mEmv2 G?m23RE?3RE?Ôò EkmEm12 ?mv?G26REmEm 3RE(2) È¡ÎÀÐÇÓëµØÇòÏà¾àÎÞÏÞÔ¶(r¡ú¡Þ)ʱµÄÊÆÄÜΪÁã,Ôò´¦ÔÚ¹ìµÀÉϵÄÎÀÐÇËù¾ßÓеÄÊÆÄÜΪ
EP??G(3) ÎÀÐǵĻúеÄÜΪ
E?Ek?EP?GmEmmmmm?GE??GE 6RE3RE6RE3 -23 Èçͼ(a)Ëùʾ,ÌìÎĹ۲ą̂ÓÐÒ»°ë¾¶ÎªR µÄ°ëÇòÐÎÎÝÃæ,ÓÐÒ»±ù¿é´Ó¹â»¬ÎÝÃæµÄ×î¸ßµãÓɾ²Ö¹ÑØÎÝÃ滬ÏÂ,ÈôĦ²ÁÁ¦ÂÔÈ¥²»¼Æ£®Çó´Ë±ù¿éÀ뿪ÎÝÃæµÄλÖÃÒÔ¼°ÔÚ¸ÃλÖõÄËٶȣ®
Ìâ 3-23 ͼ
·ÖÎö È¡±ù¿é¡¢ÎÝÃæºÍµØÇòΪϵͳ,ÓÉÓÚÎÝÃæ¶Ô±ù¿éµÄÖ§³ÖÁ¦FNʼÖÕÓë±ù¿éÔ˶¯µÄ·½Ïò´¹Ö±,¹ÊÖ§³ÖÁ¦²»×÷¹¦£»¶øÖØÁ¦PÓÖÊDZ£ÊØÄÚÁ¦,ËùÒÔ,ϵͳµÄ»úеÄÜÊغ㣮µ«ÊÇ,½öÓÐÒ»¸ö»úеÄÜÊغ㷽³Ì²»Äܽâ³öËٶȺÍλÖÃÁ½¸öÎïÀíÁ¿£»Òò´Ë,»¹ÐèÉè·¨¸ù¾Ý±ù¿éÔÚÍÑÀëÎÝÃæʱ֧³ÖÁ¦ÎªÁãÕâÒ»Ìõ¼þ,ÓÉÅ£¶Ù¶¨ÂÉÁгö±ù¿éÑؾ¶ÏòµÄ¶¯Á¦Ñ§·½³Ì£®Çó½âÉÏÊöÁ½·½³Ì¼´¿ÉµÃ³ö½á¹û£® ½â ÓÉϵͳµÄ»úеÄÜÊغã,ÓÐ
mgR?¸ù¾ÝÅ£¶Ù¶¨ÂÉ,±ù¿éÑؾ¶ÏòµÄ¶¯Á¦Ñ§·½³ÌΪ
12mv?mgRcos¦È (1) 2mv2mgRcos¦È?FN?R±ù¿éÍÑÀëÇòÃæʱ,Ö§³ÖÁ¦FN £½0,ÓÉʽ(1)¡¢(2)¿ÉµÃ±ù¿éµÄ½ÇλÖÃ
(2)
2¦È?arccos?48.2o
3±ù¿é´ËʱµÄËÙÂÊΪ
41
v?gRcos¦È?v µÄ·½ÏòÓëÖØÁ¦P ·½ÏòµÄ¼Ð½ÇΪ
2Rg 3¦Á£½90¡ã - ¦È £½41.8¡ã
3 -24 ÈçͼËùʾ,°ÑÖÊÁ¿m £½0.20 kg µÄСÇò·ÅÔÚλÖÃA ʱ,µ¯»É±»Ñ¹Ëõ¦¤l £½7.5 ¡Á10 m£®È»ºóÔÚµ¯»Éµ¯ÐÔÁ¦µÄ×÷ÓÃÏÂ,СÇò´ÓλÖÃA Óɾ²Ö¹±»ÊÍ·Å,СÇòÑعìµÀABCD Ô˶¯£®Ð¡ÇòÓë¹ìµÀ¼äµÄĦ²Á²»¼Æ£®ÒÑÖª
-2
BCDÊǰ뾶r £½0.15 m µÄ°ëÔ²»¡,AB Ïà¾àΪ2r£®Ç󵯻ɾ¢¶ÈϵÊýµÄ×îСֵ£®
Ìâ 3-24 ͼ
·ÖÎö ÈôȡСÇò¡¢µ¯»ÉºÍµØÇòΪϵͳ,СÇòÔÚ±»ÊͷźóµÄÔ˶¯¹ý³ÌÖÐ,Ö»ÓÐÖØÁ¦ºÍµ¯Á¦ÕâÁ½¸ö±£ÊØÄÚÁ¦×÷¹¦,¹ìµÀ¶ÔÇòµÄÖ§³ÖÁ¦²»×÷¹¦,Òò´Ë,ÔÚÔ˶¯µÄ¹ý³ÌÖÐ,ϵͳµÄ»úеÄÜÊغ㣮ÔËÓÃÊغ㶨ÂɽâÌâʱ,¹Ø¼üÔÚÓÚÑ¡ºÃϵͳµÄ³õ̬ºÍÖÕ̬£®Îª»ñÈ¡±¾ÌâËùÇóµÄ½á¹û,³õ̬ѡÔÚѹËõµ¯»É¸Õ±»ÊÍ·Åʱ¿Ì,ÕâÑù,¿Éʹµ¯»ÉµÄ¾¢¶ÈϵÊýÓë³õ̬ÏàÁªÏµ£»¶øÖÕ̬ÔòÈ¡ÔÚСÇò¸ÕºÃÄÜͨ¹ý°ëÔ²»¡Ê±µÄ×î¸ßµãC ´¦,ÒòΪÕâʱСÇòµÄËÙÂÊÕý´¦ÓÚÒ»ÖÖÁÙ½ç״̬,Èô´óÓÚ¡¢µÈÓÚ´ËËÙÂÊʱ,СÇò¶¨ÄÜÑعìµÀ¼ÌÐøÏòÇ°Ô˶¯£»Ð¡ÓÚ´ËËÙÂÊʱ,СÇò½«ÍÑÀë¹ìµÀÅ׳ö£®¸ÃËÙÂÊÔò¿É¸ù¾ÝÖØÁ¦ÌṩԲ»¡Ô˶¯ÖÐËùÐèµÄÏòÐÄÁ¦,ÓÉÅ£¶Ù¶¨ÂÉÇó³ö£®ÕâÑù,ÔÙÓÉϵͳµÄ»úеÄÜÊغ㶨Âɼ´¿É½â³ö¸Ãµ¯»É¾¢¶ÈϵÊýµÄ×îСֵ£®
½â СÇòÒª¸ÕºÃͨ¹ý×î¸ßµãC ʱ,¹ìµÀ¶ÔСÇòÖ§³ÖÁ¦FN £½0,Òò´Ë,ÓÐ
2mvcmg?r (1)
ȡСÇò¿ªÊ¼Ê±ËùÔÚλÖÃA ΪÖØÁ¦ÊÆÄܵÄÁãµã,ÓÉϵͳµÄ»úеÄÜÊغ㶨ÂÉ,ÓÐ
1122 (2) k?¦¤l??mg?3r??mvc22ÓÉʽ(1)¡¢(2)¿ÉµÃ
k?7mgr?366N?m?1 2?¦¤l?3 -25 ÈçͼËùʾ,ÖÊÁ¿Îªm¡¢ËÙ¶ÈΪv µÄ¸ÖÇò,ÉäÏòÖÊÁ¿Îªm¡äµÄ°Ð,°ÐÖÐÐÄÓÐһС¿×,ÄÚÓо¢¶ÈϵÊýΪk µÄµ¯»É,´Ë°Ð×î³õ´¦ÓÚ¾²Ö¹×´Ì¬,µ«¿ÉÔÚˮƽÃæÉÏ×÷ÎÞĦ²Á»¬¶¯£®Çó×Óµ¯ÉäÈë°ÐÄÚµ¯»Éºó,µ¯»ÉµÄ×î´óѹËõ¾àÀ룮
42
Ìâ 3-25 ͼ
·ÖÎö ÕâÒ²ÊÇÒ»ÖÖÅöײÎÊÌ⣮ÅöײµÄÈ«¹ý³ÌÊÇָСÇò¸ÕÓ뵯»É½Ó´¥Ö±ÖÁµ¯»É±»Ñ¹Ëõµ½×î´ó,СÇòÓë°Ð¸ÕºÃµ½´ï¹²Í¬ËÙ¶ÈΪֹ,ÔÚÕâ¹ý³ÌÖÐ,СÇòºÍ°Ð×é³ÉµÄϵͳÔÚˮƽ·½Ïò²»ÊÜÍâÁ¦×÷ÓÃ,ÍâÁ¦µÄ³åÁ¿ÎªÁã,Òò´Ë,ÔÚ´Ë·½Ïò¶¯Á¿Êغ㣮µ«ÊÇ,½ö¿¿¶¯Á¿Êغ㶨ÂÉ»¹²»ÄÜÇó³ö½á¹ûÀ´£®ÓÖ¿¼Âǵ½ÎÞÍâÁ¦¶Ôϵͳ×÷¹¦,ϵͳÎ޷DZ£ÊØÄÚÁ¦×÷¹¦,¹ÊϵͳµÄ»úеÄÜÒ²Êغ㣮ӦÓÃÉÏÊöÁ½¸öÊغ㶨ÂÉ,²¢¿¼Âǵ½ÇòÓë°Ð¾ßÓÐÏàͬËÙ¶Èʱ,µ¯»É±»Ñ¹ËõÁ¿×î´óÕâÒ»Ìõ¼þ,¼´¿ÉÇó½â£®Ó¦ÓÃÊغ㶨ÂÉÇó½â,¿ÉÃâ³ýÅöײÖеÄÐí¶àϸ½ÚÎÊÌ⣮ ½â É赯»ÉµÄ×î´óѹËõÁ¿Îªx0 £®Ð¡ÇòÓë°Ð¹²Í¬Ô˶¯µÄËÙ¶ÈΪv1 £®Óɶ¯Á¿Êغ㶨ÂÉ,ÓÐ
mv??m?m??v1 (1)
ÓÖÓÉ»úеÄÜÊغ㶨ÂÉ,ÓÐ
121122 (2) mv??m?m??v1?kx0222ÓÉʽ(1)¡¢(2)¿ÉµÃ
x0?mm?v ?k?m?m?3 -26 ÖÊÁ¿Îªm µÄµ¯ÍèA,´©¹ýÈçͼËùʾµÄ°Ú´¸Bºó,ËÙÂÊÓÉv ¼õÉÙµ½v /2£®ÒÑÖª°Ú´¸µÄÖÊÁ¿Îªm¡ä,°ÚÏß³¤¶ÈΪl,Èç¹û°Ú´¸ÄÜÔÚ´¹Ö±Æ½ÃæÄÚÍê³ÉÒ»¸öÍêÈ«µÄÔ²ÖÜÔ˶¯,µ¯ÍèËÙ¶ÈvµÄ×îСֵӦΪ¶àÉÙ£¿
Ìâ 3-26 ͼ
·ÖÎö ¸ÃÌâ¿É·ÖÁ½¸ö¹ý³Ì·ÖÎö£®Ê×ÏÈÊǵ¯Íè´©Ô½°Ú´¸µÄ¹ý³Ì£®¾Íµ¯ÍèÓë°Ú´¸Ëù×é³ÉµÄϵͳ¶øÑÔ,ÓÉÓÚ´©Ô½¹ý³ÌµÄʱ¼äºÜ¶Ì,ÖØÁ¦ºÍµÄÕÅÁ¦ÔÚˮƽ·½ÏòµÄ³åÁ¿Ô¶Ð¡ÓÚ³å»÷Á¦µÄ³åÁ¿,Òò´Ë,¿ÉÈÏΪϵͳÔÚˮƽ·½Ïò²»ÊÜÍâÁ¦µÄ³åÁ¿×÷ÓÃ,ϵͳÔڸ÷½ÏòÉÏÂú×㶯Á¿Êغ㣮°Ú´¸ÔÚÅöײÖлñµÃÁËÒ»¶¨µÄËÙ¶È,Òò¶ø¾ßÓÐÒ»¶¨µÄ¶¯ÄÜ,Ϊʹ°Ú´¸ÄÜÔÚ´¹Ö±Æ½ÃæÄÚ×÷Ô²ÖÜÔ˶¯,±ØÐëʹ°Ú´¸ÔÚ×î¸ßµã´¦ÓÐÈ·¶¨µÄËÙÂÊ,¸ÃËÙÂÊ¿ÉÓÉÆä±¾ÉíµÄÖØÁ¦ÌṩԲÖÜÔ˶¯ËùÐèµÄÏòÐÄÁ¦À´È·¶¨£»Óë´Ëͬʱ,°Ú´¸ÔÚ×÷Ô²ÖÜÔ˶¯¹ý³ÌÖÐ,°Ú´¸ÓëµØÇò×é³ÉµÄϵͳÂú×ã»úеÄÜÊغ㶨ÂÉ,¸ù¾ÝÁ½Êغ㶨Âɼ´¿É½â³ö½á¹û£® ½â ÓÉˮƽ·½ÏòµÄ¶¯Á¿Êغ㶨ÂÉ,ÓÐ
vmv?m?m?v? (1)
2 43
Ϊʹ°Ú´¸Ç¡ºÃÄÜÔÚ´¹Ö±Æ½ÃæÄÚ×÷Ô²ÖÜÔ˶¯,ÔÚ×î¸ßµãʱ,°ÚÏßÖеÄÕÅÁ¦F£Ô£½0,Ôò
2m?v?hm?g?l (2)
ʽÖÐv¡äh Ϊ°Ú´¸ÔÚÔ²ÖÜ×î¸ßµãµÄÔ˶¯ËÙÂÊ£®
ÓÖ°Ú´¸ÔÚ´¹Ö±Æ½ÃæÄÚ×÷Ô²ÖÜÔ˶¯µÄ¹ý³ÌÖÐ,Âú×ã»úеÄÜÊغ㶨ÂÉ,¹ÊÓÐ
112m?v??2m?gl?m?v?h (3) 22½âÉÏÊöÈý¸ö·½³Ì,¿ÉµÃµ¯ÍèËùÐèËÙÂʵÄ×îСֵΪ
v?2m?m5gl
3 -27 Á½ÖÊÁ¿ÏàͬµÄÎïÌå·¢ÉúÅöײ£¬ÒÑÖªÅöײǰÁ½ÎïÌåËٶȷֱðΪ£º?Ϊ?v0iºÍv0j£¬ÅöײºóÒ»ÎïÌåËÙ¶È
v0i£¬Ç󣺣¨1£©ÅöײºóÁíÒ»ÎïÌåµÄËÙ¶Èv£»£¨2£©ÅöײÖÐÁ½ÎïÌåËðʧµÄ»úеÄܹ²Îª¶àÉÙ£¿
·ÖÎö ±¾Ìâ¿ÉÖ±½ÓÔËÓö¯Á¿Êغ㶨ÂɵÄʸÁ¿Ê½Çó½â£¬ÓÉÓÚ²»ÊÇÍêÈ«µ¯ÐÔÅöײ£¬±Ø¶¨Óв¿·Ö»úеÄÜת»¯ÎªÁ½ÎïÌåµÄÄÚÄÜ.
½â £¨1£©Óɶ¯ÄÜÊغãµÃ
?mv0i?mv0j??mÅöײºóÁíÒ»ÎïÌåËÙ¶ÈΪ
v0i?mv 2v??ͨ¹ýÉÏʽ£¬¶ÁÕß»¹¿ÉÇóµÃËٶȴóСºÍ·½Ïò.
(2) ÅöײºóÁíÒ»ÎïÌåËٶȴóСΪ
v0i?v0j 2v?(?v0252)?v0?v0 2211v1212?E?[mv2?m(0)2]?(mv0?mv0)22222Ôò 12??mv04¡°-¡±ºÅ±íʾÅöײºóϵͳ»úеÄܼõÉÙÁË.
3 -28 ÖÊÁ¿Îª7.2 ¡Á10 kg,ËÙÂÊΪ6.0 ¡Á10 m¡¤s µÄÁ£×ÓA,ÓëÁíÒ»¸öÖÊÁ¿ÎªÆäÒ»°ë¶ø¾²Ö¹µÄÁ£×ÓB ·¢Éú¶þάÍêÈ«µ¯ÐÔÅöײ,ÅöײºóÁ£×ÓA µÄËÙÂÊΪ5.0 ¡Á10 m¡¤s£®Çó£º(1) Á£×ÓB µÄËÙÂʼ°Ïà¶ÔÁ£×ÓA ÔÀ´Ëٶȷ½ÏòµÄƫת½Ç£»(2) Á£×ÓA µÄƫת½Ç£®
7
-1
-23
7
-1
44
Ìâ 3-28 ͼ
·ÖÎö ÕâÊÇÁ£×ÓϵͳµÄ¶þάµ¯ÐÔÅöײÎÊÌ⣮ÕâÀàÎÊÌâͨ³£²ÉÓÃÊغ㶨ÂÉÀ´½â¾ö£®ÒòΪÁ£×ÓϵͳÔÚÅöײµÄƽÃæÄÚ²»ÊÜÍâÁ¦×÷ÓÃ,ͬʱ,ÅöײÓÖÊÇÍêÈ«µ¯ÐÔµÄ,¹ÊϵͳͬʱÂú×㶯Á¿ÊغãºÍ»úеÄÜÊغ㣮ÓÉÁ½Êغ㶨ÂÉ·½³Ì¼´¿É½âµÃ½á¹û£®
½â È¡ÈçͼËùʾµÄ×ø±ê,ÓÉÓÚÁ£×ÓϵͳÊôÓÚбÅö,ÔÚÅöײƽÃæÄÚ¸ù¾Ýϵͳ¶¯Á¿Êغ㶨ÂÉ¿ÉÈ¡Á½¸ö·ÖÁ¿Ê½,ÓÐ
1mmvA?vBcos¦Â?mv?Acos¦Á (1) 22m0?vBsin¦Â?mv?Asin¦Á (2)
2ÓÖÓÉ»úеÄÜÊغ㶨ÂÉ,ÓÐ
121?m?212mvA???vB?v?A (3) 22?2?2½âʽ(1)¡¢(2)¡¢(3)¿ÉµÃÅöײºóB Á£×ÓµÄËÙÂÊΪ
27?1? vB?2v2?v?4.69?10m?sAA??¸÷Á£×ÓÏà¶ÔÔÁ£×Ó·½ÏòµÄÆ«½Ç·Ö±ðΪ
?2v2A?3vA¦Á?arccos?22o20?
4vAv?A
¦Â?arccos3vB?54o6? 4vA3 -29 ÈçͼËùʾ,Ò»ÖÊÁ¿Îªm¡äµÄÎï¿é·ÅÖÃÔÚбÃæµÄ×îµ×¶ËA ´¦,бÃæµÄÇã½ÇΪ¦Á,¸ß¶ÈΪh,Îï¿éÓëбÃæµÄ¶¯Ä¦²ÁÒòÊýΪ¦Ì,½ñÓÐÒ»ÖÊÁ¿Îªm µÄ×Óµ¯ÒÔËÙ¶Èv0 ÑØˮƽ·½ÏòÉäÈëÎï¿é²¢ÁôÔÚÆäÖÐ,ÇÒʹÎï¿éÑØбÃæÏòÉÏ»¬¶¯£®ÇóÎï¿é»¬³ö¶¥¶ËʱµÄËٶȴóС£®
Ìâ 3-29 ͼ
·ÖÎö ¸ÃÌâ¿É·ÖÁ½¸ö½×¶ÎÀ´ÌÖÂÛ,Ê×ÏÈÊÇ×Óµ¯ºÍÎï¿éµÄײ»÷¹ý³Ì,È»ºóÊÇÎï¿é(°üº¬×Óµ¯)ÑØбÃæÏòÉϵĻ¬¶¯
45
¹ý³Ì£®ÔÚײ»÷¹ý³ÌÖÐ,¶ÔÎï¿éºÍ×Óµ¯×é³ÉµÄϵͳ¶øÑÔ,ÓÉÓÚײ»÷Ç°ºóµÄ×ܶ¯Á¿Ã÷ÏÔÊDz»Í¬µÄ,Òò´Ë,ײ»÷¹ý³ÌÖж¯Á¿
²»Êغ㣮Ӧ¸Ã×¢Òâ,²»ÊÇÈκÎÅöײ¹ý³ÌÖж¯Á¿¶¼ÊÇÊغãµÄ£®µ«ÊÇ,ÈôÈ¡ÑØбÃæµÄ·½Ïò,Òòײ»÷Á¦(ÊôÓÚÄÚÁ¦)Ô¶´óÓÚ×Óµ¯µÄÖØÁ¦P1 ºÍÎï¿éµÄÖØÁ¦P2 ÔÚбÃæµÄ·½ÏòÉϵķÖÁ¦ÒÔ¼°Îï¿éËùÊܵÄĦ²ÁÁ¦F£æ ,Ôڸ÷½ÏòÉ϶¯Á¿Êغã,Óɴ˿ɵõ½Îï¿é±»×²»÷ºóµÄËٶȣ®ÔÚÎï¿éÑØбÃæÉÏ»¬µÄ¹ý³ÌÖÐ,Ϊ½âÌâ·½±ã,¿ÉÖØÐÂÑ¡Ôñϵͳ(¼´È¡×Óµ¯¡¢Îï¿éºÍµØÇòΪϵͳ),´Ëϵͳ²»ÊÜÍâÁ¦×÷ÓÃ,¶ø·Ç±£ÊØÄÚÁ¦ÖнöĦ²ÁÁ¦×÷¹¦,¸ù¾ÝϵͳµÄ¹¦ÄÜÔÀí,¿É½âµÃ×îÖյĽá¹û£®
½â ÔÚ×Óµ¯ÓëÎï¿éµÄײ»÷¹ý³ÌÖÐ,ÔÚÑØбÃæµÄ·½ÏòÉÏ,¸ù¾Ý¶¯Á¿ÊغãÓÐ
mv0cos¦Á??m?m??v1 (1)
ÔÚÎï¿éÉÏ»¬µÄ¹ý³ÌÖÐ,ÈôÁîÎï¿é¸Õ»¬³öбÃ涥¶ËʱµÄËÙ¶ÈΪv2 ,²¢È¡A µãµÄÖØÁ¦ÊÆÄÜΪÁ㣮ÓÉϵͳµÄ¹¦ÄÜÔÀí¿ÉµÃ
?¦Ì?m?m??gcos¦Áh sin¦Á?ÓÉʽ(1)¡¢(2)¿ÉµÃ
1122 (2) ?m?m??v2??m?m??gh??m?m??v1222?m?v2??v0cos¦Á??2gh?¦Ìcot¦Á?1?
?m?m??3 -30 ÈçͼËùʾ,Ò»¸öÖÊÁ¿Îªm µÄСÇò,´ÓÄÚ±ÚΪ°ëÇòÐεÄÈÝÆ÷±ßÔµµãA»¬Ï£®ÉèÈÝÆ÷ÖÊÁ¿Îªm¡ä,°ë¾¶ÎªR,Äڱڹ⻬,²¢·ÅÖÃÔÚĦ²Á¿ÉÒÔºöÂÔµÄˮƽ×ÀÃæÉÏ£®¿ªÊ¼Ê±Ð¡ÇòºÍÈÝÆ÷¶¼´¦ÓÚ¾²Ö¹×´Ì¬£®µ±Ð¡ÇòÑØÄÚ±Ú»¬µ½ÈÝÆ÷µ×²¿µÄµãBʱ,Êܵ½ÏòÉϵÄÖ§³ÖÁ¦Îª¶à´ó£¿
Ìâ 3-30 ͼ
·ÖÎö ÓÉÓÚ×ÀÃæÎÞĦ²Á,ÈÝÆ÷¿ÉÒÔÔÚˮƽ×ÀÃæÉÏ»¬¶¯,µ±Ð¡ÇòÑØÈÝÆ÷ÄÚ±ÚÏ»¬Ê±,ÈÝÆ÷ÔÚ×ÀÃæÉÏÒ²Òª·¢ÉúÒƶ¯£®½«Ð¡ÇòÓëÈÝÆ÷ÊÓΪϵͳ,¸ÃϵͳÔÚÔ˶¯¹ý³ÌÖÐÑØˮƽ×ÀÃæ·½Ïò²»ÊÜÍâÁ¦×÷ÓÃ,ϵͳÔڸ÷½ÏòÉϵĶ¯Á¿Êغ㣻Èô½«Ð¡Çò¡¢ÈÝÆ÷ÓëµØÇòÊÓΪϵͳ,ÒòϵͳÎÞÍâÁ¦×÷ÓÃ,¶øÄÚÁ¦ÖÐÖØÁ¦ÊDZ£ÊØÁ¦,¶øÖ§³ÖÁ¦²»×÷¹¦,ϵͳµÄ»úеÄÜÊغ㣮ÓÉÁ½¸öÊغ㶨ÂɿɽâµÃСÇòºÍÈÝÆ÷ÔÚ¹ßÐÔϵÖеÄËٶȣ®ÓÉÓÚÏà¶ÔÔ˶¯µÄ´æÔÚ,СÇòÏà¶ÔÈÝÆ÷Ô˶¯µÄ¹ì¼£ÊÇÔ²,¶øÏà¶Ô×ÀÃæÔ˶¯µÄ¹ì¼£¾Í²»ÔÙÊÇÔ²ÁË,Òò´Ë,ÔÚÔËÓÃÇúÏßÔ˶¯Öеķ¨Ïò¶¯Á¦Ñ§·½³ÌÇó½âСÇòÊÜÁ¦Ê±,±ØÐë×¢Òâ²Î¿¼ÏµµÄÑ¡Ôñ£®ÈôÈ¡ÈÝÆ÷Ϊ²Î¿¼Ïµ(·Ç¹ßÐÔϵ),СÇòÔڴ˲ο¼ÏµÖеĹ켣ÈÔÊÇÈÝÆ÷Ô²»¡,Æä·¨Ïò¼ÓËٶȿÉÓɴ˿̵ÄËÙ¶È(Ïà¶ÔÓÚÈÝÆ÷ËÙ¶È)ÇóµÃ£®ÔÚ·ÖÎöСÇòÊÜÁ¦Ê±,³ýÖØÁ¦ºÍÖ§³ÖÁ¦Íâ,»¹±ØÐë¼Æ¼°ËüËùÊܵĹßÐÔÁ¦£®Ð¡ÇòλÓÚÈÝÆ÷µÄµ×²¿ÕâÒ»ÌØÊâλÖÃʱ,ÈÝÆ÷µÄ¼ÓËÙ¶ÈΪÁã,¹ßÐÔÁ¦Ò²ÎªÁ㣮ÕâÑù,ÓÉ·¨Ïò¶¯Á¦Ñ§·½³ÌÇó½âСÇòËùÊܵÄÖ§³ÖÁ¦¾ÍºÜÈÝÒ×ÁË£®ÈôÈÔÈ¡µØÃæΪ²Î¿¼Ïµ(¹ßÐÔϵ),ËäÈ»ÎÞÐ迼ÂǹßÐÔÁ¦,µ«ÊÇÒòСÇòµÄ¹ì¼£·½³Ì±È½Ï¸´ÔÓ,ÆäÇúÂʰ뾶¼°·¨Ïò¼ÓËÙ¶ÈÄÑÒÔÈ·¶¨,ʹÇó½â½ÏΪÀ§ÄÑ£®
46
½â ¸ù¾Ýˮƽ·½Ïò¶¯Á¿Êغ㶨ÂÉÒÔ¼°Ð¡ÇòÔÚÏ»¬¹ý³ÌÖлúеÄÜÊغ㶨ÂÉ¿É·Ö±ðµÃ
mvm?m?vm??0 (1)
1212mvv?m?vm??mgR (2) 22ʽÖÐvm ¡¢vm¡ä·Ö±ð±íʾСÇò¡¢ÈÝÆ÷Ïà¶Ô×ÀÃæµÄËٶȣ®ÓÉʽ(1)¡¢(2)¿ÉµÃСÇòµ½´ïÈÝÆ÷µ×²¿Ê±Ð¡Çò¡¢ÈÝÆ÷µÄËٶȴóС·Ö±ðΪ
vm?2m?gR
m?m?
vm??m2m?gRm?m?m?ÓÉÓÚСÇòÏà¶ÔµØÃæÔ˶¯µÄ¹ì¼£±È½Ï¸´ÔÓ,Ϊ´Ë,¿É¸ÄΪÒÔÈÝÆ÷Ϊ²Î¿¼Ïµ(·Ç¹ßÐÔϵ)£®ÔÚÈÝÆ÷µ×²¿Ê±,СÇòÏà¶ÔÈÝÆ÷µÄÔ˶¯ËÙ¶ÈΪ
?m?m??vm??vm???vm???vm?vm????2gR (3)
??m?ÔÚÈÝÆ÷µ×²¿,СÇòËùÊܹßÐÔÁ¦ÎªÁã,Æä·¨ÏòÔ˶¯·½³ÌΪ
2mv?mFN?mg?R (4)
ÓÉʽ(3)¡¢(4)¿ÉµÃСÇò´ËʱËùÊܵ½µÄÖ§³ÖÁ¦Îª
µÚËÄÕ ¸ÕÌåµÄת¶¯
4£1 ÓÐÁ½¸öÁ¦×÷ÓÃÔÚÒ»¸öÓй̶¨×ªÖáµÄ¸ÕÌåÉÏ£º
(1)ÕâÁ½¸öÁ¦¶¼Æ½ÐÐÓÚÖá×÷ÓÃʱ£¬ËüÃǶÔÖáµÄºÏÁ¦¾ØÒ»¶¨ÊÇÁ㣻 (2)ÕâÁ½¸öÁ¦¶¼´¹Ö±ÓÚÖá×÷ÓÃʱ£¬ËüÃǶÔÖáµÄºÏÁ¦¾Ø¿ÉÄÜÊÇÁ㣻 (3)µ±ÕâÁ½¸öÁ¦µÄºÏÁ¦ÎªÁãʱ£¬ËüÃǶÔÖáµÄºÏÁ¦¾ØÒ²Ò»¶¨ÊÇÁ㣻 (4)µ±ÕâÁ½¸öÁ¦¶ÔÖáµÄºÏÁ¦¾ØΪÁãʱ£¬ËüÃǵĺÏÁ¦Ò²Ò»¶¨ÊÇÁ㣮 ¶ÔÉÏÊö˵·¨ÏÂÊöÅжÏÕýÈ·µÄÊÇ( )
(A)Ö»ÓÐ(1)ÊÇÕýÈ·µÄ (B)(1)¡¢(2)ÕýÈ·£¬(3)¡¢(4)´íÎó (C) (1)¡¢(2)¡¢(3)¶¼ÕýÈ·£¬(4)´íÎó (D)(1)¡¢(2)¡¢(3)¡¢(4)¶¼ÕýÈ·
·ÖÎöÓë½â Á¦¶ÔÖáÖ®Á¦¾Øͨ³£ÓÐÈýÖÖÇé¿ö£ºÆäÖÐÁ½ÖÖÇé¿öÏÂÁ¦¾ØΪÁ㣺һÊÇÁ¦µÄ×÷ÓÃÏßͨ¹ýתÖᣬ¶þÊÇÁ¦Æ½ÐÐÓÚתÖá(ÀýÈçÃŵÄÖØÁ¦²¢²»ÄÜʹÃÅת)£®²»Âú×ãÉÏÊöÇé¿öϵÄ×÷ÓÃÁ¦(º¬ÌâÊö×÷ÓÃÁ¦´¹Ö±ÓÚתÖáµÄÇé¿ö)¶ÔÖáÖ®¾Ø²»ÎªÁ㣬µ«Í¬Ê±ÓÐÁ½¸öÁ¦×÷ÓÃʱ£¬Ö»ÒªÂú×ãÁ½Á¦¾Ø´óСÏàµÈ£¬·½ÏòÏà·´£¬Á½Á¦¾Ø¶ÔͬһÖáµÄºÏÍâÁ¦¾ØÒ²¿ÉÒÔΪÁ㣬ÓÉÒÔÉϹæÔò¿ÉÖª(1)(2)˵·¨ÊÇÕýÈ·£®¶ÔÓÚ(3)(4)Á½ÖÖ˵·¨£¬Èç×÷ÓÃÓÚ¸ÕÌåÉϵÄÁ½¸öÁ¦Îª¹²µãÁ¦£¬µ±ºÏÁ¦ÎªÁãʱ£¬ËüÃǶÔͬһÖáµÄºÏÍâÁ¦¾ØÒ²Ò»¶¨ÎªÁ㣬·´Ö®ÒàÈ»£®µ«ÈçÕâÁ½¸öÁ¦Îª·Ç¹²µãÁ¦£¬ÔòÒÔÉϽáÂÛ²»³ÉÁ¢£¬¹Ê(3)(4)˵·¨²»ÍêÈ«ÕýÈ·£®×ÛÉÏËùÊö£¬Ó¦Ñ¡(B)£® 4£2 ¹ØÓÚÁ¦¾ØÓÐÒÔϼ¸ÖÖ˵·¨£º
47
(1)¶Ôij¸ö¶¨Öáת¶¯¸ÕÌå¶øÑÔ£¬ÄÚÁ¦¾Ø²»»á¸Ä±ä¸ÕÌåµÄ½Ç¼ÓËٶȣ» (2)Ò»¶Ô×÷ÓÃÁ¦ºÍ·´×÷ÓÃÁ¦¶ÔͬһÖáµÄÁ¦¾ØÖ®ºÍ±ØΪÁ㣻
(3)ÖÊÁ¿ÏàµÈ£¬ÐÎ×´ºÍ´óС²»Í¬µÄÁ½¸ö¸ÕÌ壬ÔÚÏàͬÁ¦¾ØµÄ×÷ÓÃÏ£¬ËüÃǵÄÔ˶¯×´Ì¬Ò»¶¨Ïàͬ£® ¶ÔÉÏÊö˵·¨ÏÂÊöÅжÏÕýÈ·µÄÊÇ( )
(A)Ö»ÓÐ(2)ÊÇÕýÈ·µÄ (B)(1)¡¢(2)ÊÇÕýÈ·µÄ
(C)(2)¡¢(3)ÊÇÕýÈ·µÄ (D)(1)¡¢(2)¡¢(3)¶¼ÊÇÕýÈ·µÄ
·ÖÎöÓë½â ¸ÕÌåÖÐÏàÁÚÖÊÔªÖ®¼äµÄÒ»¶ÔÄÚÁ¦ÊôÓÚ×÷ÓÃÁ¦Óë·´×÷ÓÃÁ¦£¬ÇÒ×÷ÓõãÏàͬ£¬¹Ê¶ÔͬһÖáµÄÁ¦¾ØÖ®ºÍ±ØΪÁ㣬Òò´Ë¿ÉÍÆÖª¸ÕÌåÖÐËùÓÐÄÚÁ¦¾ØÖ®ºÍΪÁ㣬Òò¶ø²»»áÓ°Ïì¸ÕÌåµÄ½Ç¼ÓËٶȻò½Ç¶¯Á¿µÈ£¬¹Ê(1)(2)˵·¨ÕýÈ·£®¶Ô˵·¨(3)À´Ëµ£¬ÌâÊöÇé¿öÖÐÁ½¸ö¸ÕÌå¶ÔͬһÖáµÄת¶¯¹ßÁ¿ÒòÐÎ×´¡¢´óС²»Í¬ÓпÉÄܲ»Í¬£¬Òò¶øÔÚÏàͬÁ¦¾Ø×÷ÓÃÏ£¬²úÉúµÄ½Ç¼ÓËٶȲ»Ò»¶¨Ïàͬ£¬Òò¶øÔ˶¯×´Ì¬Î´±ØÏàͬ£¬Óɴ˿ɼûӦѡ(B)£®
4£3 ¾ùÔÈϸ°ôOA¿ÉÈÆͨ¹ýÆäÒ»¶ËO¶øÓë°ô´¹Ö±µÄˮƽ¹Ì¶¨¹â»¬Öáת¶¯£¬ÈçͼËùʾ£¬½ñʹ°ô´ÓˮƽλÖÃÓɾ²Ö¹¿ªÊ¼×ÔÓÉÏÂÂ䣬ÔÚ°ô°Úµ½ÊúֱλÖõĹý³ÌÖУ¬ÏÂÊö˵·¨ÕýÈ·µÄÊÇ( ) (A)½ÇËٶȴÓСµ½´ó£¬½Ç¼ÓËٶȲ»±ä (B)½ÇËٶȴÓСµ½´ó£¬½Ç¼ÓËٶȴÓСµ½´ó (C)½ÇËٶȴÓСµ½´ó£¬½Ç¼ÓËٶȴӴóµ½Ð¡ (D)½ÇËٶȲ»±ä£¬½Ç¼ÓËÙ¶ÈΪÁã
·ÖÎöÓë½â ÈçͼËùʾ£¬ÔÚ°ôÏÂÂä¹ý³ÌÖУ¬ÖØÁ¦¶ÔÖáÖ®¾ØÊDZ仯µÄ£¬Æä´óСÓë°ôºÍˮƽÃæµÄ¼Ð½ÇÓйأ®µ±°ô´¦ÓÚˮƽλÖã¬ÖØÁ¦¾Ø×î´ó£¬µ±°ô´¦ÓÚÊúֱλÖÃʱ£¬ÖØÁ¦¾ØΪÁ㣮Òò´ËÔÚ°ôÔÚÏÂÂä¹ý³ÌÖÐÖØÁ¦¾ØÓÉ´óµ½Ð¡£¬ÓÉת¶¯¶¨ÂÉÖª£¬°ôµÄ½Ç¼ÓËÙÒàÓÉ´óµ½Ð¡£¬¶ø°ôµÄ½ÇËÙ¶ÈÈ´ÓÉСµ½´ó(ÓÉ»úеÄÜÊغãÒà¿ÉÅжϽÇËٶȱ仯Çé¿ö)£¬Ó¦Ñ¡(C)£®
4£4 Ò»Ô²ÅÌÈÆͨ¹ýÅÌÐÄÇÒ´¹Ö±ÓÚÅÌÃæµÄˮƽÖáת¶¯£¬Öá¼äĦ²Á²»¼Æ£®ÈçͼÉäÀ´Á½¸öÖÊÁ¿Ïàͬ£¬ËٶȴóСÏàͬ£¬·½ÏòÏà·´²¢ÔÚÒ»ÌõÖ±ÏßÉϵÄ×Óµ¯£¬ËüÃÇͬʱÉäÈëÔ²Å̲¢ÇÒÁôÔÚÅÌÄÚ£¬Ôò×Óµ¯ÉäÈëºóµÄ˲¼ä£¬Ô²Å̺Í×Óµ¯ÏµÍ³µÄ½Ç¶¯Á¿LÒÔ¼°Ô²Å̵ĽÇËٶȦصı仯Çé¿öΪ( ) (A)L ²»±ä£¬¦ØÔö´ó (B)Á½Õß¾ù²»±ä (C)L²»±ä£¬¦Ø¼õС (D)Á½Õß¾ù²»È·¶¨
·ÖÎöÓë½â ¶ÔÓÚÔ²ÅÌÒ»×Óµ¯ÏµÍ³À´Ëµ£¬²¢ÎÞÍâÁ¦¾Ø×÷Ó㬹Êϵͳ¶ÔÖáO µÄ½Ç¶¯Á¿Êغ㣬¹ÊL²»±ä£¬´ËʱӦÓÐÏÂʽ³ÉÁ¢£¬¼´
mvd?mvd?J0¦Ø0?J¦Ø
ʽÖÐmvdΪ×Óµ¯¶ÔµãOµÄ½Ç¶¯Á¿?0ΪԲÅ̳õʼ½ÇËٶȣ¬J Ϊ×Óµ¯ÁôÔÚÅÌÖкóϵͳ¶ÔÖáOµÄת¶¯¹ßÁ¿£¬J0Ϊ×Ó
48
µ¯ÉäÈëÇ°Å̶ÔÖáOµÄת¶¯¹ßÁ¿£®ÓÉÓÚJ£¾J0£¬Ôò?£¼?0£®¹ÊÑ¡(C)£®
4£5 ¼ÙÉèÎÀÐÇ»·ÈƵØÇòÖÐÐÄ×÷ÍÖÔ²Ô˶¯£¬ÔòÔÚÔ˶¯¹ý³ÌÖУ¬ÎÀÐǶԵØÇòÖÐÐĵÄ( ) (A)½Ç¶¯Á¿Êغ㣬¶¯ÄÜÊغã (B)½Ç¶¯Á¿Êغ㣬»úеÄÜÊغã (C)½Ç¶¯Á¿²»Êغ㣬»úеÄÜÊغã (D)½Ç¶¯Á¿²»Êغ㣬¶¯Á¿Ò²²»Êغã (£Å)½Ç¶¯Á¿Êغ㣬¶¯Á¿Ò²Êغã
·ÖÎöÓë½â ÓÉÓÚÎÀÐÇÒ»Ö±Êܵ½ÍòÓÐÒýÁ¦×÷Ó㬹ÊÆ䶯Á¿²»¿ÉÄÜÊغ㣬µ«ÓÉÓÚÍòÓÐÒýÁ¦Ò»Ö±Ö¸ÏòµØÇòÖÐÐÄ£¬ÔòÍòÓÐÒýÁ¦¶ÔµØÇòÖÐÐĵÄÁ¦¾ØΪÁ㣬¹ÊÎÀÐǶԵØÇòÖÐÐĵĽǶ¯ÐÇÊغ㣬¼´r ¡Ámv£½ºãÁ¿£¬Ê½ÖÐrΪµØÇòÖÐÐÄÖ¸ÏòÎÀÐǵÄλʸ£®µ±ÎÀÐÇ´¦ÓÚÍÖÔ²¹ìµÀÉϲ»Í¬Î»ÖÃʱ£¬ÓÉÓÚ£ür £ü²»Í¬£¬ÓɽǶ¯Á¿ÊغãÖªÎÀÐÇËÙÂʲ»Í¬£¬ÆäÖе±ÎÀÐÇ´¦ÓÚ½üµØµãʱËÙÂÊ×î´ó£¬´¦ÓÚÔ¶µØµãʱËÙÂÊ×îС£¬¹ÊÎÀÐǶ¯Äܲ¢²»Êغ㣬µ«ÓÉÍòÓÐÒýÁ¦Îª±£ÊØÁ¦£¬ÔòÎÀÐǵĻúеÄÜÊغ㣬¼´ÎÀÐǶ¯ÄÜÓëÍòÓÐÒýÁ¦ÊÆÄÜÖ®ºÍά³Ö²»±ä£¬Óɴ˿ɼû£¬Ó¦Ñ¡(B)£®
4£6 Ò»Æû³µ·¢¶¯»úÇúÖáµÄתËÙÔÚ12 s ÄÚÓÉ1.2¡Á10r¡¤min¾ùÔȵÄÔö¼Óµ½2.7¡Á10r¡¤min£®(1)ÇóÇúÖáת¶¯µÄ½Ç¼ÓËٶȣ»(2)ÔÚ´Ëʱ¼äÄÚ£¬ÇúÖáתÁ˶àÉÙת£¿
·ÖÎö ÕâÊǸÕÌåµÄÔ˶¯Ñ§ÎÊÌ⣮¸ÕÌ嶨Öáת¶¯µÄÔ˶¯Ñ§¹æÂÉÓëÖʵãµÄÔ˶¯Ñ§¹æÂÉÓÐÀàËƵĹØϵ£¬±¾ÌâΪÔȱäËÙת¶¯£®
½â (1)ÓÉÓÚ½ÇËٶȦأ½2¦Ðn(n Ϊµ¥Î»Ê±¼äÄÚµÄתÊý)£¬¸ù¾Ý½Ç¼ÓËٶȵĶ¨Òå¦Á½Ç¼ÓËÙ¶ÈΪ
3
-1
3
-1
?d¦Ø£¬ÔÚÔȱäËÙת¶¯ÖÐdt¦Á?(2)·¢¶¯»úÇúÖáת¹ýµÄ½Ç¶ÈΪ
¦Ø?¦Ø02¦Ð?n?n0???13.1rad?s?2
tt???0t??t2?ÔÚ12 sÄÚÇúÖáת¹ýµÄȦÊýΪ
12???02t?¦Ð?n?n0?t
N?¦Èn?n0?t?390Ȧ 2¦Ð2-47
2
4£7 Ë®·Ö×ÓµÄÐÎ×´ÈçͼËùʾ£¬´Ó¹âÆ×·ÖÎö֪ˮ·Ö×Ó¶ÔAA¡äÖáµÄת¶¯¹ßÁ¿JAA¡ä£½1.93 ¡Á10kg¡¤m£¬¶ÔBB¡äÖáת¶¯¹ßÁ¿JBB¡ä£½1.14 ¡Á10kg¡¤m£¬ÊÔÓÉ´ËÊý¾ÝºÍ¸÷Ô×ÓÖÊÁ¿Çó³öÇâºÍÑõÔ×ӵľàÀëDºÍ¼Ð½Ç¦È£®¼ÙÉè¸÷Ô×Ó¶¼¿Éµ±Öʵ㴦Àí£®
-47
2
49
Ìâ 4-7 ͼ
·ÖÎö È罫Ô×ÓÊÓΪÖʵ㣬ÔòË®·Ö×ÓÖеÄÑõÔ×Ó¶ÔAA¡äÖáºÍBB¡äÖáµÄת¶¯¹ßÁ¿¾ùΪÁ㣬Òò´Ë¼ÆËãË®·Ö×Ó¶ÔÁ½¸öÖáµÄת¶¯¹ßÁ¿Ê±£¬Ö»Ð迼ÂÇÇâÔ×Ó¼´¿É£® ½â ÓÉͼ¿ÉµÃ
JAA??2mHd2sin2¦È JBB??2mHd2cos2¦È
´Ë¶þʽÏà¼Ó£¬¿ÉµÃJAA?Ôò d?JBB??2mHd2
?JAA??JBB??9.59?10?11m
2mH?tan2¦È
ÓɶþʽÏà±È£¬¿ÉµÃ JAA?/JBB?Ôò ¦È?arctanJAA?1.93?arctan?52.3o JBB?1.144£8 Ò»·ÉÂÖÓÉÒ»Ö±¾¶Îª30©M£¬ºñ¶ÈΪ2.0©MµÄÔ²Å̺ÍÁ½¸öÖ±¾¶Îª10©M£¬³¤Îª8.0©MµÄ¹²ÖáÔ²ÖùÌå×é³É£¬Éè·ÉÂÖµÄÃܶÈΪ7.8¡Á10kg¡¤m£¬Çó·ÉÂÖ¶ÔÖáµÄת¶¯¹ßÁ¿£®
3
-3
Ìâ 4-8 ͼ
·ÖÎö ¸ù¾Ýת¶¯¹ßÁ¿µÄ¿Éµþ¼ÓÐÔ£¬·ÉÂÖ¶ÔÖáµÄת¶¯¹ßÁ¿¿ÉÊÓΪԲÅÌÓëÁ½Ô²ÖùÌå¶ÔͬÖáµÄת¶¯¹ßÁ¿Ö®ºÍ£»¶øÔÈÖÊÔ²ÅÌ¡¢Ô²ÖùÌå¶ÔÖáµÄת¶¯¹ßÁ¿µÄ¼ÆËã¿É²éÊéÖй«Ê½£¬»ò¸ù¾Ýת¶¯¹ßÁ¿µÄ¶¨Ò壬Óüòµ¥µÄ»ý·Ö¼ÆËãµÃµ½£® ½â ¸ù¾Ýת¶¯¹ßÁ¿µÄµþ¼ÓÐÔ£¬ÓÉÔÈÖÊÔ²ÅÌ¡¢Ô²ÖùÌå¶ÔÖáµÄת¶¯¹ßÁ¿¹«Ê½¿ÉµÃ
50